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Unit 6 - Simplifying Radicals Introduction

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Unit 6 - Simplifying Radicals Introduction

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Using exponent rules to simplify radicals or square roots

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Unit 6 - Simplifying Radicals with variables

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Unit 6 - Simplifying Radicals with variables

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Converting from radical to exponential expression: two variables

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Sal shows how to rewrite ∜(5a^4b^12) as an exponential expression and then simplify it.

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Unit 6 - Introduction to Complex Numbers

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Unit 6 - Introduction to Complex Numbers

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Sal explains how we obtain complex numbers by adding real numbers and imaginary numbers.

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Unit 6 - Simplifying Square Roots of Negative Numbers using i

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Unit 6 - Simplifying Square Roots of Negative Numbers using i

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We're asked to simplify the principal square root of negative 52. And we're going to assume, because we have a negative 52 here inside of the radical, that this is the principal branch of the complex square root function. That we can actually put, input, negative numbers in the domain of this function. That we can actually get imaginary, or complex, results. So we can rewrite negative 52 as negative 1 times 52. So this can be rewritten as the principal square root of negative 1 times 52. And then, if we assume that this is the principal branch of the complex square root function, we can rewrite this. This is going to be equal to the square root of negative 1 times-- or I should say, the principal square root of negative 1 times the principal square root of 52. Now, I want to be very, very clear here. You can do what we just did. If we have the principal square root of the product of two things, we can rewrite that as the principal square root of each, and then we take the product. But you can only do this, or I should say, you can only do this if either both of these numbers are positive, or only one of them is negative. You cannot do this if both of these were negative. For example, you could not do this. You could not say the principal square root of 52 is equal to negative 1 times negative 52. You could do this. So far, I haven't said anything wrong. 52 is definitely negative 1 times negative 52. But then, since these are both negative, you cannot then say that this is equal to the square root of negative 1 times the square root of negative 52. In fact, I invite you to continue on this train of reasoning. You're going to get a nonsensical answer. This is not OK. You cannot do this, right over here. And the reason why you cannot do this is that this property does not work when both of these numbers are negative. Now with that said, we can do it if only one of them are negative or both of them are positive, obviously. Now, the principal square root of negative 1, if we're talking about the principal branch of the complex square root function, is i. So this right over here does simplify to i. And then let's think if we can simplify the square root of 52 any. And to do that, we can think about its prime factorization, see if we have any perfect squares sitting in there. So 52 is 2 times 26, and 26 is 2 times 13. So we have 2 times 2 there, or 4 there, which is a perfect square. So we can rewrite this as equal to-- Well, we have our i, now. The principal square root of negative 1 is i. The other square root of negative 1 is negative i. But the principal square root of negative 1 is i. And then we're going to multiply that times the square root of 4 times 13. And this is going to be equal to i times the square root of 4. i times the square root of 4, or the principal square root of 4 times the principal square root of 13. The principal square root of 4 is 2. So this all simplifies, and we can switch the order, over here. This is equal to 2 times the square root of 13. 2 times the principal square root of 13, I should say, times i. And I just switched around the order. It makes it a little bit easier to read if I put the i after the numbers over here. But I'm just multiplying i times 2 times the square root of 13. That's the same thing as multiplying 2 times the principal square root of 13 times i. And I think this is about as simplified as we can get here.

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We're asked to simplify the principal square root of negative 52. And we're going to assume, because we have a negative 52 here inside of the radical, that this is the principal branch of the complex square root function. That we can actually put, input, negative numbers in the domain of this function.

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Unit 6 - Introduction to imaginary numbers

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Unit 6 - Introduction to imaginary numbers

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Sal introduces the imaginary unit i, which is defined by the equation i^2=-1. He then gets to know this special number better by thinking about its powers.

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Unit 6 - Solving Square Root Equations Example 2

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Unit 6 - Solving Square Root Equations Example 2

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- [Voiceover] What I want to do in this video is get a little bit of experience, see a few examples of trying to roughly estimate the square root of non-perfect squares. So let's say that I had, if I wanted to estimate the square root of 32. And in particular, I'm just curious, between what two integers will this square root lie? Well one way to think about it is 32 is in between what perfect squares? We see 32 is, actually let me make sure I have some space for future examples. So 32, what's the perfect square below 32? So the greatest perfect square below 32 is 25. 32 is greater than 25. That's five squared. So maybe I should write it this way. So five squared is less than 32 and then 32, what's the next perfect square after 32? Well 32 is less than 36. So we could say 32 is less than six squared. So if you were to take the square root of all of these sides right over here, we could say that instead of here we have all of the values squared, but instead, if we took the square root, we could say five is going to be less than the square root of 32, which is less than, which is less than six. Notice, to go from here to here, to go from here to here, and here to here, all we did is we squared things, we raised everything to the second power. But the inequality should still hold. So the square root of 32 should be between five and six. It's going to be five point something. Let's do another example. Let's say we wanted to estimate, we want to say between what two integers is the square root of 55? Well we can do the same idea. Let's square it. So if we square the square root of 55, we're just gonna get to 55. We're just going to get, let me do that in the same color, 55. So okay, 55 is between which two perfect squares? So the perfect square that is below 55, or I could say the greatest perfect square that is less than 55. Let's see, six squared is 36 and seven squared is 49, eight squared is 64. So it would be 49. I could write that as seven squared. Let me write that, that is the same thing as seven squared. And what's the next perfect square above it? Well we just figured it out. Seven squared is 49, eight squared is larger than 55, it's 64. So this is going to be less than 64, which is eight squared. And of course 55, just to make it clear what's going on. 55 is the square root of 55 squared. That's kind of by definition, it's going to be the square root of 55 squared. And so the square root of 55 is going to be between what? It's going to be between seven and eight. So seven is less than the square root of 55, which is less than eight. So once again, this is just an interesting way to think about, what would you, if someone said the square root of 55 and at first you're like, "Oh, uh, I don't know what that is. "I don't have a calculator," et cetera et cetera. You're like, "Oh wait, wait, that's going to be between "49 and 64, so it's going to be seven point something." It's going to be seven point something. And you can even get a rough estimate of seven point what based on how far away it is from 49 and 64. You can begin to approximate things. Let's do one more example. Let's say we wanted to figure out where does the square root of 123 lie? And like always, I encourage you to pause the video and try to think about it yourself. Between what two integers does this lie? Well, if we were to square it, you get to 123. And what's the perfect square that is the greatest perfect square less than 123? Let's see, 10 squared is 100. 11 squared is 121. 12 squared is 144. So 11 squared. So 123, so we could write 121 is less than 123, which is less than 144, that's 12 squared. So if we take the square roots we could write that 11 is less than the square root of 123, which is less than 144. So once again, what's the square root of 123? It's going to be 11 point something. And in fact, it's going to be closer to 11 than it's going to be to 12. 123 is a lot closer to 121 than it is to 144. So it might be, I don't know, 11.1, something like that. I don't know if that's exactly right, we would have to check that on the calculator. But hopefully this gives you, oops I, that actually will be less than 144. But if we want to think about what consecutive integers is that be between, it's going to be a 12 right over there. Almost made a... Well anyway, you get the idea. Hopefully you enjoyed that.

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Learn how to find the approximate values of square roots. The examples used in this video are √32, √55, and √123. The technique used is to compare the squares of whole numbers to the number we're taking the square root of.

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Sal manipulates expressions like 1/√2 and 12/(2-√5) so there's no radical in the denominator.

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Unit 6 - Solving Square Root Equations Example

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Unit 6 - Solving Square Root Equations Example

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Unit 6 - Introduction to Square Root Equations

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Unit 6 - Introduction to Square Root Equations

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Intro to square-root equations & extraneous solutions

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Sal explains what square-root equations are, and shows an example of solving such an equation and checking for extraneous solutions.

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Sal checks if the ordered pairs (3,5) and (1,-7) are solutions of the inequality 5x-3y≥25.

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Graphing two variable linear inequalities

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Solving Absolut Value Inequalities

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Explains how to solve, graph, and create absolute-value inequalities.

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Solving Absolute Value Equations

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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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Schedule for Inequalities Unit

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Factoring Difference of Cubes

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Sal factors 40c^3-5d^3 as 5(2x-d)(4c^2+2cd+d^2) using a special product form for a difference of cubes.

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Factoring Sum of Cubes

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Sal factors 27x^6+125 as (3x^2+5)(9x^4-15x^2+25) using a special product form for a sum of cubes.

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Simplifying Rational Expressions (intro)

**Click "View Original" in the top left of the pop up. Then, check out the other videos (listed on the left to see more examples of this)

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Intro to rational expression simplification

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Sal explains what it means to simplify a rational expression and why we would want to do that. Just don't forget the excluded values!

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Factoring Example 3.5

* More Factoring Quadratics (easy version)

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More examples of factoring quadratics as (x+a)(x+b)

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Can't get enough of Sal factoring simple quadratics? Here's a handful of examples just for you!

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Factoring Example 8

*Difference of Squares (harder)

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Factoring difference of squares: leading coefficient ≠ 1

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Factoring Example 7

*Difference of Squares (introduction)

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Factoring difference of squares: two variables

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Factoring Example 6

* Factoring by taking GCF and grouping

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Factoring quadratics: common factor + grouping

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Factoring Example 5

*Factoring by Grouping (#2)

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Factoring Example 4

*Factoring by Grouping (introduction)

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Sal factors t^2+8t+15 as (t+3)(t+5), first using the "sum-product" method, then using the grouping method.

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Factoring Example 3

*Factoring Quadratics (easy version)

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Factoring quadratics as (x+a)(x+b) (example 2)

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Factoring Example 2

* Factoring out the GCF in a trinomial

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Example 1: Factoring trinomials with a common factor

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Factoring trinomials with a common factor

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Factoring Example 1

*Factoring out the GCF from a linear binomial

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Factoring with the distributive property

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Sal shows how to factor the expression 4x+18 into the expression 2(2x+9).