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Volume of cones

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Volume of cylinders

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Volume of spheres

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What are the exponent rules and why are
exponents useful in doing mathematics?

Why is it useful to put numbers into the
form of Scientific Notation?

What does it mean for something to
increase exponentially?

How are the properties of integer exponents derived?

Why is scientific notation needed?

How is the volume formula applied in real life
mathematical situations?

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Students will use the meaning of exponents to multiply and divide numerical expressions with integer or fractional base2).

Students will use the meaning of exponents to raise a simple numerical expression with integer or fractional base to a power.

Students will convert a fractional or integer base that is raised to a negative exponent into an expression with a positive exponent.

Students will use the exponential rules to simplify complex expressions where they need to multiply or divide like bases, or raise a power to a power when some of the exponents are negative.

Students use the rules of exponents to solve word problems.

Students will use their knowledge of exponents to write a number in scientific notation.

Students will use their knowledge of exponents to convert a number in scientific notation into standard form.

Students will determine if a number in scientific notation describes a very large or very small quantity.

Given two numbers in scientific notation, students will determine how many more times one number
is from the other.

Students will be able to shift the decimal and change the exponent on the ten, in order to make a number that is not in scientific notation into one that is. For example, into (we
will call this a decimal shift).

Students will multiply numbers in scientific notation including ones where the results need have a decimal shift.

Students will divide numbers in scientific notation including ones where the results require a decimal shift.

Students add or subtract numbers in scientific notation that have the same exponents of 10.

Students add or subtract numbers in scientific notation that have different powers of 10. This means that they require a decimal shift so that they will have the same exponent of 10 before adding or subtracting.

Students will know the volume formula for cones, cylinder, and sphere.

Student will use their knowledge of volume to find the volumes of complex figures.

Students will be able to apply these volume formulas into real world word problems.

Students will use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number.

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PRIORITY:
EE.1​ Know and apply the properties of integer exponents to generate equivalent numerical
expressions. For example, 3² × 3⁻⁵ = 3–3      = 1/33      = 1/27.

EE.3​ Use numbers expressed in the form of a single digit times an integer power of 10 to
estimate very large or very small quantities, and to express how many times as much one is
than the other. For example, estimate the population of the United States as 3 × 10⁸ and the
population of the world as 7 × 10⁹, and determine that the world population is more than 20
times larger.

EE.4​ Perform operations with numbers expressed in scientific notations, including problems
where both decimal and scientific notation are used. Use scientific notation and choose units
of appropriate size for measurements of very large or very small quantities. (e.g., use
millimeters per year for seafloor spreading). Interpret scientific notation that has been
generated by technology.

G.9​ Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.

SUPPORTING:
EE.2:​ Use square root and cube root symbols to represent solutions to equations of the form x²= p and x³= p, where p is a positive rational number. Evaluate square roots of small perfect
squares and cube roots of small perfect cubes. Know that is irrational.

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Simplify
Apply
Conclude
Represent
Formula
Expand
Notation
Calculate
Compute

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Cubic
Squared
Cubed
Solutions
Exponent
Base
Power
Coefficient
Constant
Variable
Expression
Equation
Magnitude
Scientific notation
Base
Power
Volume
Area
Perimeter
Pi
Radius
Cone
Cylinder
Sphere

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Scientific Notation worksheet- from the video

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WCSD's Mrs. Peck discusses the many uses of scientific notation. Learn how to convert numbers to scientific notation and standard decimal notation.

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Scientific Notation

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Scientific notation intro

Scientific notation intro

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Scientific notation (practice) | Khan Academy

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Adding/Subtracting in Scientific Notation

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Adding & subtracting in scientific notation (practice) | Khan Academy

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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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Multiplying numbers in Scientific Notation

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Multiplying & dividing in scientific notation

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Scientific Notation challenge

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Practice solving addition, subtraction, multiplication, and division problems involving scientific notation.

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- [Voiceover] I would like to multiply nine times 10 to the ninth power times negative two, times ten to the negative third power. And so I encourage you, pause the video, see if you can work through this. Alright. So the first thing I would want to do is let's just change the order of multiplication. Let's multiply the nine and the negative two first and then we can multiply the 10 to the ninth and 10 to the negative third power. So if I were to change the order, we could write this as nine times, nine times, nine times negative two. Nine times negative two times 10 to the ninth power. Times ten, let me do it in that, in this color. Times ten to the ninth power. Times 10 to the negative third power. Times 10 to the negative third power. Now what's nine times negative two? Well, if nine times two is 18, nine times negative two is gonna be negative 18. So that's negative 18. And what is 10 to the ninth times 10 to the negative three? Well if I have a number raised to some exponent times that same number, I have 10, I have the same base for both of these, time that same number raised to another exponent, this is going to be the same thing as that number, as that base, raised to the sum of these exponents. This comes straight out of our exponents properties. So this is going to be 10 to the, 10 to the nine plus negative three power. Nine plus, actually, let me use those colors so you see where that nine and that negative three came from. Nine plus negative three power. Nine plus, this color, I'm having trouble changing colors. Nine plus negative three power. Now what's this gonna be? Well this is going to be equal to negative 18 times times 10 to the, nine plus negative three, same thing as nine minus three, which is six. Negative 18 times 10 to the sixth power. And if you wanted to think about well what number is this? 10 to the sixth, that's one with six zeros, this is a million. Negative 18 times a million, it's gonna be negative 18 million. Or we could say negative 18 times 10 the the sixth. Either way. But this is another way we could have written this. We could have written this as negative 18, negative 18, let me write the zeros with the green, just for fun. Negative 18. Negative 18 million. Either way is a legitimate way to represent this number.

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Multiplying multiples of powers of 10 (video) | Khan Academy

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Voiceover] I would like to multiply nine times 10 to the ninth power times negative two, times ten to the negative third power. And so I encourage you, pause the video, see if you can work through this. Alright. So the first thing I would want to do is let's just change the order of multiplication.

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Multiplication and division with powers of ten

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Multiplication and division with powers of ten

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Practice solving multiplication and division problems that involve powers of ten.

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Approximating with powers of 10

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Practice comparing two real-world quantities by approximating them with powers of 10.

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Multiply & divide powers (integer exponents)

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Exponent properties review

Exponent properties review

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Multiply & divide powers (integer exponents) (practice) | Khan Academy

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Properties of exponents challenge (integer exponents)

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Properties of exponents challenge (integer exponents)

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A quiz that cover all exponent properties using integer exponents.

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Volumes of cylinders

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Practice applying the volume formulas for cylinders.

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Volumes of cones

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Practice applying the volume formulas for cones.

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Volumes of spheres

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Practice applying the volume formulas for spheres.

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Volume of cylinders, spheres, and cones word problems

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Volume of cylinders, spheres, and cones word problems

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The formula for the volume of a cone is V=1/3hπr². Learn how to use this formula to solve an example problem.

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The formula for the volume of a sphere is V = 4/3 πr³. See the formula used in an example where we are given the diameter of the sphere.

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A cylinder's volume is π r² h, and its surface area is 2π r h + 2π r². Learn how to use these formulas to solve an example problem.

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Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers:

Like this:

Or this:

It makes it easy to use big and small values.

OK, How Does it Work?

Example: 700

Why is 700 written as 7 × 10² in Scientific Notation ?

Both 700 and have the same value, just shown in different ways.

Example: 4,900,000,000

1,000,000,000 = 10 ⁹ ,
so 4,900,000,000 = in Scientific Notation

The number is written in two parts:

Just the digits, with the decimal point placed after the first digit, followed by

× 10 to a power that puts the decimal point where it should be
(i.e. it shows how many places to move the decimal point).

In this example, 5326.6 is written as 5.3266 × 10³,
because 5326.6 = 5.3266 × 1000 = 5.3266 × 10 ³

Try It Yourself

Enter a number and see it in Scientific Notation:

Now try to use Scientific Notation yourself:

images/sci-notation-num.js

Other Ways of Writing It
3.1 × 10^8
We can use the ^ symbol (above the 6 on a keyboard), as it is easy to type.

3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000

Calculators often use "E" or "e" like this:

Example: 3.12E4 is the same as 3.12 × 10⁴

How to Do it
To figure out the power of 10, think "how many places do I move the decimal point?"
Example: 0.0055 is written 5.5 × 10⁻³

Because 0.0055 = 5.5 × 0.001 = 5.5 × 10⁻³

Example: 3.2 is written 3.2 × 10⁰

We didn't have to move the decimal point at all, so the power is

But it is now in Scientific Notation

Check!

After putting the number in Scientific Notation, just check that:

The "digits" part is between 1 and 10 (it can be 1, but never 10)

The "power" part shows exactly how many places to move the decimal point

Why Use It?

Because it makes it easier when dealing with very big or very small numbers, which are common in Scientific and Engineering work.

Example: it is easier to write (and read) 1.3 × 10⁻⁹ than 0.0000000013

It can also make calculations easier, as in this example:

Example: a tiny space inside a computer chip has been measured to be 0.00000256m wide, 0.00000014m long and 0.000275m high.
What is its volume?
Let's first convert the three lengths into scientific notation:

Then multiply the digits together (ignoring the ×10s):

2.56 × 1.4 × 2.75 = 9.856

Last, multiply the ×10s:

10⁻⁶ × 10⁻⁷ × 10⁻⁴ = 10⁻¹⁷ (easier than it looks, just add −6, −4 and −7 together)

The result is 9.856×10⁻¹⁷ m³

It is used a lot in Science:

Example: Suns, Moons and Planets

The Sun has a Mass of 1.988 × 10 ³⁰ kg.

Easier than writing 1,988,000,000,000,000,000,000,000,000,000 kg
(and that number gives a false sense of many digits of accuracy.)

It can also save space! Here is what happens when you double on each square of a chess board:

Values are rounded off, so 53,6870,912 is shown as just 5×10 ⁸

That last value, shown as 9×10 ¹⁸ is actually 9,223,372,036,854,775,808

Engineering Notation

Engineering Notation is like Scientific Notation, except that we only use powers of ten that are multiples of 3 (such as 10 ³, 10⁻³, 10 ¹² etc).

Examples:

Notice that the "digits" part can now be between 1 and 1,000 (it can be 1, but never 1,000).

The advantage is that we can replace the ×10s with Metric Numbers. So we can use standard words (such as thousand or million), prefixes (such as kilo, mega) or the symbol (k, M, etc)

Example: 19,300 meters is written 19.3 × 10³ m, or 19.3 km

Example: 0.00012 seconds is written 120 × 10⁻⁶ s, or 120 microseconds

1340, 1341, 1342, 1344, 1345, 5077, 5078, 5079, 1343, 1346, 1347, 5080

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Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers: Like this: Or this: It makes it easy to use big and small values. Why is 700 written as 7 × 102 in Scientific Notation ? Both 700 and have the same value, just shown in different ways.

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We get this amazing thing that the volume of a cone and sphere together make a cylinder (assuming they fit each other perfectly)

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Exponents are also called Powers or Indices

The exponent of a number says how many times to use the number in a multiplication.

In words: 8 ² could be called "8 to the second power", "8 to the power 2" or simply "8 squared"

Try it yourself:

So an Exponent saves us writing out lots of multiplies!

a⁷ = a × a × a × a × a × a × a = aaaaaaa

Notice how we wrote the letters together to mean multiply? We will do that a lot here.

The Key to the Laws

Writing all the letters down is the key to understanding the Laws

Which shows that x²x³ = x⁵, but more on that later!

So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.

All you need to know ...

The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas:

The exponent says how many times to use the number in a multiplication.

A negative exponent means divide, because the opposite of multiplying is dividing

If you understand those, then you understand exponents!

And all the laws below are based on those ideas.

Laws of Exponents

Here are the Laws (explanations follow):

Law

Example

x ¹ = x

6 ¹ = 6

x ⁰ = 1

7 ⁰ = 1

x⁻¹ = 1/x

4⁻¹ = 1/4

x mx n = x m+n

x ²x ³ = x ²⁺³ = x ⁵

x m/x n = x m-n

x ⁶/x ² = x ⁶⁻² = x ⁴

(x m) n = x mn

(x ²) ³ = x 2×3 = x ⁶

(xy) n = x ny n

(xy) ³ = x ³y ³

(x/y) n = x n/y n

(x/y) ² = x ² / y ²

x-n = 1/x n

x⁻³ = 1/x ³

And the law about Fractional Exponents:

Laws Explained

The first three laws above (, and ) are just part of the natural sequence of exponents. Have a look at this:

Example: Powers of 5

.. etc..

.. etc..

Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).

With x mx n, how many times do we end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.

Like the previous example, how many times do we end up multiplying "x"? Answer: "m" times, then reduce that by "n" times (because we are dividing), for a total of "m-n" times.

(Remember that x/ x = 1, so every time you see an x "above the line" and one "below the line" you can cancel them out.)

This law can also show you why x⁰=1 :

First you multiply "m" times. Then you have to do that "n" times, for a total of m×n times.

Example: (x³)⁴ = (xxx)⁴ = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x¹²

To show how this one works, just think of re-arranging all the "x"s and "y"s as in this example:

Similar to the previous example, just re-arrange the "x"s and "y"s

The law that

OK, this one is a little more complicated!

I suggest you read Fractional Exponents first, or this may not make sense.

Anyway, the important idea is that:

And so a fractional exponent like is really saying to do a cube (3) and a square root (1/2), in any order.

Just remember from fractions that m/n = m × (1/n):

Example:

The order does not matter, so it also works for m/n = (1/n) × m:

Example:

Exponents of Exponents ...

What about this example?

We do the exponent at the top first, so we calculate it this way:

And That Is It!
If you find it hard to remember all these rules, then remember this:
you can work them out when you understand the
three ideas near the top of this page

Oh, One More Thing ... What if x = 0?

The Strange Case of 0⁰

There are different arguments for the correct value of 0 ⁰

0 ⁰ could be 1, or possibly 0, so some people say it is really "indeterminate":

x ⁰ = 1, so ...

0 ⁰ = 1

0 n = 0, so ...

0 ⁰ = 0

When in doubt ...

0 ⁰ = "indeterminate"

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Exponents are also called Powers or Indices The exponent of a number says how many times to use the number in a multiplication. In words: 8 2 could be called "8 to the second power", "8 to the power 2" or simply "8 squared" Try it yourself: So an Exponent saves us writing out lots of multiplies!

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Exponent Products

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Exponents Quotients