# data[].attributes.body
I can explain the concepts of function, domain, and range.

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Functions assign a single output for each of their inputs. In this video, we see examples of various kinds of functions.

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I can evaluate and order fractions.

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I can evaluate expressions and use the order of operations with integers.

# data[].attributes.body
I understand and can use basic algebraic symbols with operations.

# data[].attributes.body
I can combine like terms.

# data[].attributes.body
I can solve a one and two step equations. (Absolute Value, Distribution, Fraction Buster, Square Root)

# data[].attributes.body
I can create equations with one variable and use them to solve problems.

# data[].attributes.body
I can rewrite an equation for the indicated variable.

# data[].attributes.body
I can use function notation and evaluate functions.

# data[].attributes.body
I can scale and graph a two variable equation.

# data[].attributes.body
I can see the connections between the four representations of a function: equation, table, graph, situation.

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Solving Simpler Absolute-Value Equations | Purplemath

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Demonstrates how to solve absolute-value equations, and explains the reasoning behind the algebra. Provides an example of how graphing can be used to confirm (or correct) results.

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Now is your chance to learn how to use the distributive property and combining like terms in order to solve more complex equations.

It seems pretty easy to learn all of these skills in isolation, but using them together to solve one problem is the key in Algebra 1.

So, how does this work? Here are a few steps to take when you come across an algebra equation that looks a little more challenging.

Steps for Solving Algebra Equations

If you see parenthesis, with more than one term inside, then distribute first!

Rewrite your equations with like terms together. Take the sign in front of each term.

Combine like terms.

Continue solving the one or two-step equation.

Let's look at a couple of examples to clarify those steps for you.

This first example is a pretty basic equation that involves the distributive property. Take note of how I distribute first before applying the rules for solving equations.

Example 1: Using the Distributive Property

That example was pretty easy, I know! Let's look at one that requires a few more steps.

This next example looks more confusing because the distributive property comes right in the middle of the equation.

You still must distribute first and then combine like terms before solving the equation.

Example 2: More Distributive Property

Ok... just one more example. This one is a little more difficult. You will have to distribute twice and you must make sure that when you distribute the 4, you actually distribute a negative 4.

Remember to take the sign in front. Pay close attention to the signs in this example.

Example 3 - Double Distributive Property!

So, how did you do? Are you ready to move onto equations with fractions?

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Using the Distributive Property When Solving Equations

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Now is your chance to learn how to use the distributive property and combining like terms in order to solve more complex equations. It seems pretty easy to learn all of these skills in isolation, but using them together to solve one problem is the key in Algebra 1.

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Fraction buster by Trisha Paul - February 12, 2013

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Solving Radical Equations: Concepts | Purplemath

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Introduces the techniques used and concepts involved in solving radical equations.

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I can calculate the rate of change for a function. I can interpret the units of the slope.

# data[].attributes.body
I can create linear equations with two variables and use them to solve real world problems.

# data[].attributes.body
I can graph a linear function and label the key features.

# data[].attributes.body
I can sketch and describe key features of a graph. (Growth, X-intercepts, Y-intercept, Max/Min, Vertex, Axis of Symmetry)

# data[].attributes.body
I can understand basic transformations to a linear function.

# data[].attributes.body
I can determine the domain and range of a linear function.

# data[].attributes.body
I can determine if a solution is viable or non-viable.

# data[].attributes.body
I can see the connections between the four representations of a linear function: equation, table, graph, situation.

# data[].attributes.body
I can graph a quadratic function and label key features.

# data[].attributes.body
I can sketch and describe key features of a graph. (Growth, X-intercepts, Y-intercept, Max/Min, Vertex, Axis of Symmetry)

# data[].attributes.body
I can rewrite a quadratic function to identify key features. (Vertex Form, Standard Form, Factored Form)

# data[].attributes.body
I can understand basic transformations to a quadratic function.

# data[].attributes.body
I can factor completely, with special cases, and by grouping. (including a>1 terms)

# data[].attributes.body
I can solve a quadratic equation using factoring and zero product property, using the quadratic formula, and by completing the square.

# data[].attributes.body
I can determine the domain and range of a quadratic function.

# data[].attributes.body
I can determine if a solution is viable or non-viable.

# data[].attributes.body
I can see the connections between the four representations of a quadratic function: equation, table, graph, situation.

# data[].attributes.body
I can apply the rules of exponents.

# data[].attributes.body
I can evaluate expressions with radicals and rational exponents.

# data[].attributes.body
I can graph an exponential function and label the key features.

# data[].attributes.body
I can sketch and describe key features of a graph. (Growth, X-intercepts, Y-intercept, Max/Min, Vertex, Axis of Symmetry)

# data[].attributes.body
I can understand basic transformations to an exponential function.

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LT 5F: Recognizing Exponential Growth/Decay

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LT 5F: Recognizing Exponential Growth/Decay

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I can identify an exponential function as growth or decay. I understand the equivalence between fraction/decimal/percent and its effect on the multiplier.

# data[].attributes.body
I can create an exponential equation given a graph, a table, or a situation and use it to solve a real world problem.

# data[].attributes.body
I can determine the domain and range of an exponential function.

# data[].attributes.body
I can determine if a solution is viable or non-viable.

# data[].attributes.body
I can see the connections between the four representations of an exponential function: equation, table, graph, situation.

# data[].attributes.body
I can create a system of equations with two variables and use them to solve problems.

# data[].attributes.body
I can solve a system of equations by graphing (correct scales and labels).

# data[].attributes.body
I can solve a system of equations by the substitution method.

# data[].attributes.body
I can solve a system of equations by the elimination method.

# data[].attributes.body
I can use technology to find the point of intersection of two graphs and explain why.

# data[].attributes.body
I can determine if there are limitations in a situation and explain them. (parallel, coinciding, no solution)

# data[].attributes.body
I can determine if a solution is viable or non-viable.

# data[].attributes.body
I can construct a viable argument to justify a solution method.

# data[].attributes.body
I can prove that an exponential function will exceed a linear function over time. (By graphing or by a table.)

# data[].attributes.body
I understand all of the symbols associated with inequalities and understand what a solution set represents. I can explain the limitations.

# data[].attributes.headline
LT 7B: Solve and Graph One-Variable Inequalities

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LT 7B: Solve and Graph One-Variable Inequalities

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I can solve inequalities with one variable. I can graph inequalities with one-variable and correctly scale it.

# data[].attributes.headline
LT 7C: Solve and Graph Two-Variable Inequalities

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LT 7C: Solve and Graph Two-Variable Inequalities

# data[].attributes.body
I can solve and graph the solutions of a linear inequality.

# data[].attributes.body
I can graph the solution set to a system of linear inequalities in two variables.

# data[].attributes.body
I can create an inequality and use it to solve real world problems.

# data[].attributes.body
I can explain that sequences are functions whose domain is a subset of integers. (Continuous vs. Discrete)

# data[].attributes.body
I can recognize whether a situation, table, graph, or equation is arithmetic (linear), geometric (exponential), or something else.

# data[].attributes.body
I can write explicit and recursive equations for arithmetic sequences.

# data[].attributes.body
I can write explicit and recursive equations for geometric sequences.

# data[].attributes.body
I can graph arithmetic and geometric sequences.

# data[].attributes.body
I can calculate mean, median and mode and make decisions and predictions with two or more sets of data.

# data[].attributes.body
I can represent data with a dot plot, a histogram, and a box plot. I can understand the distribution of the data, including the understanding of outliers.

# data[].attributes.body
I can organize categorical data in a two-way table and gather information from that data.

# data[].attributes.headline
LT 9D: Scatter Plots and Lines of Best Fit

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LT 9D: Scatter Plots and Lines of Best Fit

# data[].attributes.body
I can create a scatter plot of paired numerical data and accurately fit a line with an equation.

# data[].attributes.body
I can assess the line of best fit using the residuals.

# data[].attributes.body
I can describe the association of the data in terms of strength, direction and form. I understand that association does not imply causation. I can describe data distribution by shape. I can use the r correlation to describe the strength of the linear association.

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8-4 Multiple Representations of Functions

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Video transcript

Solve P equals 2l plus 2w for l. So this right here, this is just the formula for the perimeter of a rectangle. Perimeter is equal to 2 times the length plus 2 times the width. But they just want us to solve this equation right here, solve for l. So let's do that. So we have P is equal to 2 times l plus 2 times w. So we need to solve for l. So let's isolate all of the l terms on one side. And the best way, we could just do that by leaving it here on the right and then getting rid of this 2w. And the best way to get rid of this 2w is to subtract it from the right. But if you're going to subtract it from the right, you also have to subtract it from the left if you want this equality to hold. So you have to also subtract it from the left. And so the left-hand side becomes P minus 2w. And the right-hand side, you get-- this 2w minus 2w cancels out. You just have a 2l. And then if you want to solve for l, you just have to divide both sides of this equation by 2. You just divide both sides of this equation by 2. And we have isolated our l. We get l is equal to P minus 2w over 2. Or if we wanted to write it the other way, you could write l is equal to P minus 2w over 2. And we are done.

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Solving an equation for a variable (video) | Khan Academy

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If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

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Explains what "literal" equations are, and demonstrates how to solve them (especially formulas from geometry) for the specified variable.

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- [Voiceover] Anna wants to celebrate her birthday by eating pizza with her friends. For $42.50 total, they can buy p boxes of pizza. Each box of pizza costs $8.50. Select the equation that matches this situation. So before I even look at these, let's see if I can make sense of the sentence here. So for $42.50 total, and I'll just write 42.5, especially because in all these choices they didn't write 42.50, they just wrote 42.5 which is equivalent. So 42.50 that's the total amount they spent on pizza and if I wanted to figure out how many boxes of pizza they could buy, I could divide the total amount they spend, I could divide that by the price per box. That would give me the number of boxes. So this is the total, total dollars. This right over here is the dollar per box and then this would give me the number of boxes. # of boxes. Now other ways that I could think about it. I could say, well what's the total that they spend? So 42.50, but what's another way of thinking about the total they spend? Well you could have the amount they spend per box, times the number of boxes. So this is the total they spend and this another way of thinking about the total they spend, so these two things must be equal. So let's see, if I can see anything here that looks like this, well actually this first choice, this, is exactly, is exactly what I wrote over here. Let's see this choice right over here. P is equal to 8.5 x 42.5. Well we've already been able to write an equation that has explicitly, that has just a p on one side and so when you solve for just a p on one side, you get this thing over here, not this thing, so we could rule that out. Over here it looks kind of like this, except the p is on the wrong side. This has 8.5p is equal to 42.5, not 42.5p is equal to 8.5. If we try to get the p on the other side here, you could divide both sides by p, but then you would get p divided by p is one. You would get 42.5 is equal to 8.5/p which is not true. We have 8.5 times p is equal to 42.5, so this is, this is not going to be the case. One thing to realize, no matter what you come up with, if you came up with this first, or if you came up with this first, you can go between these two with some algebraic manipulations. So for example, to go from this blue one to what I wrote in red up here, you just divide both sides by 8.5. So you divide by 8.5 on the left, you divide 8.5 on the right. Obviously to keep the equal sign you have to do the same thing to the left and right, but now you would have 42.5/8.5 is equal to, is equal to p. Which is exactly what we have over there. Let's do one more of these. Good practice. Mr. Herman's class is selling candy for a school fundraiser. The class has a goal of raising $500 by selling c boxes of candy. For every box they sell, they make $2.75. Write an equation that the students could solve to figure out how many boxes of candy they need to sell. How many boxes of candy they need to sell. Well there's a couple of ways you could think about it. They have a goal of raising $500 and so they want to get a total of $500, and if each box is $2.75, divide the total by the amount they get each per box and then this is going to be equal to the number of boxes that they need to sell. So this we've done. This is an equation that the students could solve to figure out how many boxes of candy they need to sell. Another way you could think about it, it's 2.75 per box times c boxes. This is the total amount of money they will raise. Whoops, this is the amount. So this is the amount that they will raise and their goal is that, their goal is to raise $500. So they want this to be equal to $500. So this also could be an equation that the students could solve to figure out how many boxes of candy they need to sell.

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Voiceover] Anna wants to celebrate her birthday by eating pizza with her friends. For $42.50 total, they can buy p boxes of pizza. Each box of pizza costs $8.50. Select the equation that matches this situation. So before I even look at these, let's see if I can make sense of the sentence here.

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Let's think about what it means to multiply 2 over 3, or 2/3, times 4/5. In a previous video, we've already seen how we can actually compute this. This is going to be equal to-- in the numerator, we just multiply the numerators. So it's going to be 2 times 4. And in the denominator, we just multiply the denominator. So it's going to be 3 times 5. And so the numerator is going to be 8, and the denominator is going to be 15. And this is about as simple as we can make it. 8 and 15 don't have any factors common to each other, than 1, so this is what it is. It's 8/15. But how, why does that actually makes sense? And to think about it, we'll think of two ways of visualizing it. So let's draw 2/3. I'll draw it relatively big. So I'm going to draw 2/3, and I'm going to take 4/5 of it. So 2/3, and I'm going to make it pretty big. Just like this. So this is 1/3. And then this would be 2/3. Which I could do a little bit better job making those equal, or at least closer to looking equal. So there you go. I have thirds. Let me do it one more time. So here I have drawn thirds. 2/3 represents 2 of them. It represents 2 of them. One way to think about this is 2/3 times 4/5 is 4/5 of this 2/3. So how do we divide this 2/3 into fifths? Well, what if we divided each of these sections into 5. So let's do that. So let's divide each into 5. 1, 2, 3, 4, 5. 1, 2, 3, 4, 5. And I could even divide this into 5 if I want. 1, 2, 3, 4, 5. And we want to take 4/5 of this section here. So how many fifths do we have here? We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And we've got to be careful. These really aren't fifths. These are actually 15ths, because the whole is this thing over here. So I should really say how many 15ths do we have? And that's where we get this number from. But you see if 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. Where did that come from? I had 3, I had thirds. And then I took each of those thirds, and I split them into fifths. So then I have five times as many sections. 3 times 5 is 15. But now we want 4/5 of this right over here. This is 10/15 right over here. Notice it's the same thing as 2/3. Now if we want to take 4/5 of that, if you have 10 of something, that's going to be 8 of them. So we're going to take 8 of them. So 1, 2, 3, 4, 5, 6, 7, 8. We took 8 of the 15, so that is 8/15. You could have thought about it the other way around. You could have started with fifths. So let me draw it that way. So let me draw a whole. So this is a whole. Let me cut it into five equal pieces, or as close as I can draw five equal pieces. 1, 2, 3, 4, 5. 4/5, we're going to shade in 4 of them. 4 of the 5 equal pieces. 3, 4. And now we want to take 2/3 of that. Well, how can we do that? Well, let's split each of these 5 into 3 pieces. So now we have essentially 15ths again. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. We want to take 2/3 of this yellow area. We're not taking 2/3 of the whole section. We're taking 2/3 of the 4/5. So how many 15ths do we have here? We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So if you have 12 of something, and you want to take 2/3 of that, you're going to be taking 8 of it. So you're going to be taking 1, 2, 3, 4, 5, 6, 7, 8 or 8 of the fifteenths now. So either way, you get to the same result. One way, you're thinking of taking 4/5 of 2/3. Another way you could think of it as you're taking 2/3 of 4/5.

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Intro to multiplying 2 fractions (video) | Khan Academy

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Let's think about what it means to multiply 2 over 3, or 2/3, times 4/5. In a previous video, we've already seen how we can actually compute this. This is going to be equal to-- in the numerator, we just multiply the numerators. So it's going to be 2 times 4.

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Video transcript

let's think about what it means to take 8/3 and divide it by one-third so let me draw a number line here so there is my number line this is 0 this is 1 and this is 2 maybe this is 3 right over here and let me plot 8/3 so to do that I just need to break up my each each hole into thirds so let's see it's one-third two-thirds three-thirds 4/3 5/3 6 thirds 7/3 8/3 so right over here and then of course 9/3 would get us to 3 so this right over here is 8/3 8/3 now one way to think about 8/3 divided by 3 is what if we take this length and we say how many jumps would it take to get there if we're doing it in jumps of 1/3 or essentially we're breaking this up if we were to break up 8/3 into sections of 1/3 how many sections would I have or how many jumps what I have well let's think about that if we're trying to take jumps of 1/3 we're gonna have to go 1 2 3 4 5 6 7 8 jumps so we could view this as we could you let me do this in a different color in this orange so we took these 8 jumps right over here so we could view 8/3 divided by 1/3 as being equal to 8 now why is this actually make sense well when you're dividing things in 2/3 for every hole you're now going to have 3 jumps so whatever value you're trying to get to you're going to have that number times 3 jumps so another way of thinking about it is that 8/3 8/3 divided by divided by 1/3 is the same thing as 8/3 8/3 times 3 now we could either write it like this we could write x 3 like that or if we want to write 3 as a fraction we know that 3 is the same thing three over one and when we already know how to multiply fractions a multiply the numerators eight times three so you have eight let me do that that same color you have eight times three in the numerator now eight times three and then you have three times 1 in the denominator three times three times one in the denominator which would give you 24 thirds which is the same thing as 24 divided by three which once again is equal is equal to is equal to is equal to eight now let's see if this still makes sense instead of dividing by one-third if we were divided by two-thirds so let's think about what 8/3 divided by two-thirds is divided by two-thirds well once again this is like asking the question if we wanted to break up this section from zero to 8/3 into sections of 2/3 or jumps of two-thirds how many sections or how many jumps would I have to make well let's think about it one jump just in a different color we could make one jump that's the same color as my 8/3 we could do one jump my computer is doing something strange we could do one jump two jumps three jumps and four jumps so we see 8/3 divided by 2/3 is equal is equal to four now does this make sense in this world right over here well if we take 8/3 and we do the same thing saying hey look dividing by a fraction is the same thing as multiplying by a reciprocal well let's multiply by three halves let's multiply by the reciprocal of 2/3 so we swap the numerator and the denominator so we multiply it times three halves and then what do we get in the numerator once again we get eight times three which is 24 and in the denominator we get 3 times 2 which is 6 so now we get 24 divided by 6 is equal to four now does it make sense that we got half the answer if you think about the difference between what we did here and what we did here these are almost the same except here we really just didn't divide or you could say you divided by one while here you / - well does that make sense well sure because here you jumped twice as far you jumped twice as far so you had to take half the number of steps and so in the first example you saw why it makes sense to multiply by 3 when you divide by a fraction for every hole you're making 3 jumps so that's why when you divide by this fraction you multiply or when you divide whatever's in the denominator you multiply by it and now when the numerator is greater than 1 every jump you're going twice as far as you did in this first one right over here and so you would have to do half as many jumps hopefully that makes sense it's easy to think about just mechanically how to divide fractions dividing 8/3 divided by 1/3 is the same thing as 8/3 times 3 over 1 or 8/3 divided by 2/3 is the same thing as 8/3 times 3 over 2 but hopefully this this video gives you a little bit more of an intuition of why this is the case

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Understanding division of fractions (video) | Khan Academy

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If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

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Order of Operations presented by Math Goodies. Learn order of operations with our self-paced instruction.

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The fundamental concept behind the order of operations is to perform arithmetic operators in the "right" order or sequence. Let's take a look at how Rob and Patty tried to simplify a given numerical expression by applying the order or rule of operations.

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Intro to order of operations (video) | Khan Academy

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This example shows the steps and clarifies the purpose of order of operations: to have ONE way to interpret a mathematical statement.

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Symbols save time and space when writing. Here are the most common algebraic symbols:

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Example: Calculate the square root of 10 ( ) to 2 decimal places.

1. Find the two perfect square numbers it lies between.

Solution:3
² = 9 and 4 ² = 16, so lies between 3 and 4.

2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer)

3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667

Repeat step 2: 10/3.1667 = 3.1579
Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623

Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001

If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.

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Example: Calculate the square root of 10 () to 2 decimal places. 1. Find the two perfect square numbers it lies between. Solution:3 2 = 9 and 4 2 = 16, so lies between 3 and 4. 2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer) 3.

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First learn about Squares, then Square Roots are easy.

How to Square A Number

To square a number: multiply it by itself.

Example: What is 3 squared?

3 Squared

=

= 3 × 3 = 9

"Squared" is often written as a little 2 like this:

This says"4 Squared equals 16"(the little 2 says the number appears twice in multiplying)

Negative Numbers

We can also square negative numbers.

That was interesting!

When we square a negative number we get a positive result.

Just the same as squaring a positive number:

(For more detail read Squares and Square Roots in Algebra)

Square Roots

A square root goes the other way:

3 squared is 9, so a square root of 9 is 3

A square root of a number is ...

... a value that can be multiplied by itself to give the original number.

A square root of 9 is ...

It is like asking:

What can we multiply by itself to get this?

To help you remember think of the root of a tree:

"I know the tree, but what root made it?"

In this case the tree is "9", and the root is "3".

Here are some more squares and square roots:

4

16

5

25

36

49

Decimal Numbers

It also works for decimal numbers.

Try the sliders below (note: '...' means the decimals continue on forever):

Using the sliders:

Negatives

We discovered earlier that we can square negative numbers:

Example: (−3) squared

(−3) × (−3) = 9

And of course 3 × 3 = 9 also.

So the square root of 9 could be −3 or +3

Example: What are the square roots of 25?

(−5) × (−5) = 25

5 × 5 = 25

So the square roots of 25 are −5 and +5

The Square Root Symbol

This is the special symbol that means "square root", it is sort of like a tick,
and actually started hundreds of years ago as a dot with a flick upwards.

It is called theradical, and always makes mathematics look important!

We use it like this:

and we say"square root of 9 equals 3"

Example: What is √25?

25 = 5 × 5, in other words when we multiply 5 by itself (5 × 5) we get 25
So the answer is:
√25 = 5

But wait a minute! Can't the square root also be −5? Because (−5) × (−5) = 25 too.

Well the square root of 25 could be −5 or +5.

But when we use the radical symbol √ we only give the positive (or zero) result.

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Perfect Squares

The Perfect Squares (also called "Square Numbers") are the squares of the integers:

Try to remember them up to 12.

Calculating Square Roots

It is easy to work out the square root of a perfect square, but it is really hard to work out other square roots.

Example: what is √10?

Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.

Getting closer to 10, but it will take a long time to get a good answer!

At this point, I get out my calculator and it says:

3.1622776601683793319988935444327

But the digits just go on and on, without any pattern.

So even the calculator's answer is only an approximation !

Note: numbers like that are called Irrational Numbers, if you want to know more.
The Easiest Way to Calculate a Square Root

Use your calculator's square root button!

And also use your common sense to make sure you have the right answer.

A Fun Way to Calculate a Square Root

There is a fun method for calculating a square root that gets more and more accurate each time around:

a) start with a guess (let's guess 4 is the square root of 10)

b) divide by the guess (10/4 = 2.5)
c) add that to the guess (4 + 2.5 = 6.5)
d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25)
e) now, set that as the new guess , and start at b) again

And so, after 3 times around the answer is 3.1623, which is pretty good, because:
3.1623 x 3.1623 = 10.00014
Now ... why don't you try calculating the square root of 2 this way?

How to Guess

What if we have to guess the square root for a difficult number such as "82,163" ... ?

In that case we could think "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 is a good start.

Square Root Day

The 4th of April 2016 is a Square Root Day, because the date looks like 4/4/16

The next after that is the 5th of May 2025 (5/5/25)

309,310,315, 1082, 1083, 2040, 3156, 2041, 2042, 3154

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First learn about Squares, then Square Roots are easy. To square a number: multiply it by itself. 3 Squared = = 3 × 3 = 9 "Squared" is often written as a little 2 like this: This says"4 Squared equals 16" (the little 2 says the number appears twice in multiplying) We can also square negative numbers.

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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How to Cube A Number

To cube a number, just use it in a multiplication 3 times ...

Example: What is 3 Cubed?

3 Cubed

=

=

3 × 3 × 3

= 27

Note: we write "3 Cubed" as (the little ³ means the number appears three times in multiplying)

Cube Root

A cube root goes the other direction:

3 cubed is 27, so the cube root of 27 is 3

3

27

The cube root of a number is ...... a special value that when
cubed gives the original number.

Note: When you see "root" think

"I know the tree, but what is the root that produced it?"

In this case the tree is "27", and the cube root is "3".

Here are some cubes and cube roots:

Example: What is the Cube root of 125?

Well, we just happen to know that 125 = 5 × 5 × 5 (if we use 5 three times in a multiplication we get 125) ...
... so the cube root of 125 is 5
The Cube Root Symbol

This is the special symbol that means "cube root", it is the "radical" symbol (used for square roots) with a little three to mean cube root.

You can use it like this:

(we say "the cube root of 27 equals 3")

You Can Also Cube Negative Numbers

Have a look at this:

So the cube root of −125 is −5

Perfect Cubes

The Perfect Cubes are the cubes of the whole numbers:

It is easy to work out the cube root of a perfect cube, but it is really hard to work out other cube roots.

Example: what is the cube root of 30?

Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.

We are getting closer, but very slowly ... at this point, I get out my calculator and it says:

... but the digits just go on and on, without any pattern. So even the calculator's answer is only an approximation !

(Further reading: these kind of numbers are calledsurds which are a special type of irrational number)

316, 2007, 2008, 5081, 5084, 3157, 3158, 5082, 5083, 5085

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To cube a number, just use it in a multiplication 3 times ... 3 Cubed = = 3 × 3 × 3 = 27 Note: we write "3 Cubed" as (the little 3 means the number appears three times in multiplying) A cube root goes the other direction: 3 cubed is 27, so the cube root of 27 is 3 3 27 The cube root of a number is ......

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Video transcript

let's say that I've got to Chuck Norris's or maybe it's Chuck nor I and to that I am going to add another three Chuck Norris's so I'm gonna add another three Chuck Norris's and this might seem a little bit obvious but how many Chuck Norrises do I now have well two Chuck Norris's we can represent this is literally a Chuck Norris plus a Chuck Norris so let me do that a Chuck Norris plus another Chuck Norris - Chuck Norris's you could also use it as two times Chuck Norris and this is just another way of representing it and three Chuck Norris is three is Chuck Norris is you could do that as a Chuck Norris plus a Chuck Norris plus another Chuck Norris and so we would have a grand total and this might be very simple for you but you would have a grand total of one two three four five Chuck Norrises so this would be equal to this would be equal to five Chuck Norrises now let's get a little bit more abstract here Chuck Norris is a very tangible thing so let's go to a little bit more of traditional algebraic notation if I have 2 X's 2 X's and rember 2x this is what we could do this as 2 X's or 2 times X and to that I were to add 3 X's and to that I would add 3 X's how many X's do I have well once again 2 X's that's 2 times X you could view that as an X plus an X we don't know what the value of x is but whatever that value is we can add it to itself and then 3 X's are going to be that value let me do the neck same green color 3 X's are going to be that value plus that value plus whatever that value is and so how many X's do I now have well I'm going to have 1 2 3 4 5 X's so 2x plus 3x is equal to is equal to 5x and if you think about it all we really did and hopefully you conceptualize added the two numbers that we're multiplying the X and these numbers the 2 or the 3 they're called coefficients very fancy word but it's just this constant number this regular number that's multiplied by the variable you just add the two and the three to get your 5x now let's think about this a little bit more let's go back to this original expression the two Chuck Norrises plus three Chuck Norris's let's say two that we were to add let's say we were to add some type of a let's say we were to add seven plums over here so this is my drawing of a plum so we have seven plums plus two Chuck Norris's plus three Chuck Norrises and let's say that I add another two plums I add another two plums here so what would this whole thing be well I wouldn't add the 7 to the 2 to the 3 plus the to the we're adding different things here you have two Chuck Norris's and three Chuck Norris's so they're still going to simplify to five Chuck Norrises and then we would separately think about the plums we have seven plums and we're adding another two plums we're going to have nine plums plus nine plums so this simplifies to five Chuck Norrises and nine plums similarly over here if I had instead of just 2x plus 3x if I had seven y 7y plus 2x plus 3x plus 2y what do i now have well I can't add the XS and the Y's they could very well represent a different number so all I can do is really add the X's and then I get the 5x and then I'd separately add the Y if I have seven wise and to that I add two y's I'm going to have nine wise if I have seven of something and I had two of something I now have nine of that something so I'm going to have nine wise so you add that we've done a different color you add this and this you get that you add the x's you add the x's you get that right over there so hopefully that makes a little sense actually I'll throw out one more idea so given this what would happen if I were to have two x plus one plus seven X plus 5 well once again you might be tempted to add the two plus the one but they're adding different things these are two x's this is just the number one so you really just have to add the x's together so you're going to say well I got two x's and I'm going to add seven X's to that well that means I had now have 9x is and then separately you would say well I've got just the abstract number one and then I got another five one plus five is going to be equal to six

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Intro to combining like terms (video) | Khan Academy

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When you were learning how to add or subtract numbers, you may have started with similar objects such as fruits. For instance, if there are three apples in a box and you place two more, how many apples are there? Obviously, it is possible to add them together because the fruits involved are of the same kind.

In algebra, this idea of adding apples can be represented using the simple equation below. Notice that we are able to combine the two terms, 3a and 2a , because they have the same variable part which is a. We say that they are similar or like terms.

Now, how about if we have apples and bananas? Can we simply add them too? The answer is NO. Apples and bananas are two different kinds of fruits, so putting them together is just creating a pile of fruits. It is not possible to say that "3 apples plus 2 bananas is equal to 5 applebananas". There's no such thing!

Examples of How to Combine Like Terms with or without the Distributive Property

Now, let's take a look at some examples!

Example 1: Simplify the expression by combining like terms.

There are four terms in this algebraic expression. Two terms with similar x-variables, and , and two constant terms which are 6 and . Since they are like terms, we can definitely combine them using the required operation.

I would place similar terms side by side first before combining them together.

Example 2: Simplify the expression by combining like terms.

I see that we have three kinds of similar terms that can be combined which are the y-terms, x-terms, and constant terms.

Let's rearrange the expression such that similar terms are adjacent or close to each other. This allows us to have a "laser" focus on which terms to add or subtract.

Example 3: Simplify the expression by combining like terms.

Don't assume that terms with the same variable are automatically like or similar terms. Remember that each variable must also have a matching exponent value. Be careful to distinguish that and are not like terms because the first x-variable has an exponent of 2, while the second is 1.

In fact we have three distinct groups of like terms here; namely, terms with , x, and the pure numbers (constants). Move around the terms in the expression such that like terms are placed side by side before we combine them either by adding or subtracting. Don't forget also to keep the original signs in front of each term when you move them around.

Just a quick note, the term - x is understood to have a coefficient of -1. Hence, .

Example 4: Simplify the expression by combining like terms.

We can't combine like terms just yet because of the presence of a parenthesis. We need to somehow get rid of that before proceeding with the usual steps. If you are not sure how to start this out, remember to replace that minus symbol in front of the parenthesis by .

The next step is to distribute that through the terms inside the parenthesis. By doing so, two things happen! The parenthesis is eliminated, and the terms inside the parenthesis will switch signs.

Now that the parenthesis is gone, we can proceed with steps in combining like terms. Group together similar terms, and then add or subtract them whatever operation is required.

Example 5: Simplify the expression .

I hope you see this one coming! This is an "upgrade" to the previous problem because this time around, the expression contains two parentheses. It should be clear at this point that we should eliminate them before we can move forward.

We begin by distributing the constant terms into the terms inside the parenthesis.

Then, rearrange the terms so that similar terms are clustered together. Finally, combine like terms by adding or subtracting whichever is required.

Example 6: Simplify the expression .

This is an example of nested grouping symbols. Observe that there is a parenthesis inside the main square bracket. We will simplify the inner parenthesis first and then work ourselves outward. Let's simplify this step by step.

Step 1: Combine the x-variables within the inner parenthesis.

Step 2: Distribute the constant 2 into the terms of the inner parenthesis.

Step 3: Combine like terms as much as possible inside the square bracket symbol. Identify like terms first before combining.

Step 4: Distribute the constant through the terms inside the square bracket to get rid of it.

Step 5: Finish this off by combining like terms again that show up after eliminating the square bracket symbol.

Example 7: Simplify the expression .

Example 8: Simplify the expression .

Example 9: Simplify the expression .

Example 10: Simplify the expression .

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When you were learning how to add or subtract numbers, you may have started with similar objects such as fruits. For instance, if there are three apples in a box and you place two more, how many apples are there? Obviously, it is possible to add them together because the fruits involved are of the same kind.

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Fractions Packet (Notes and Practice Problems with Answers!)

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Fractions Packet (Notes and Practice Problems with Answers!)

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Ex 1: Key Characteristics of the Graph of  a Quadratic Function (Opens Up)

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A one-step equation is an algebraic equation you can solve in only one step. You've solved the equation when you get the variable by itself, with no numbers in front of it, on one side of the equal sign.

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What Is Absolute Value Anyway? - Best Explanation

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Get Cluegle on Google Play https://play.google.com/store/apps/details?id=com.lovvtide.cluegle Explains absolute value in a way that actually makes sense. Using a number line, the concept of absolute value is illustrated through several examples.

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Notes and Practice