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Author:
Nate Muckley

This packet teaches the definitions, uses, and properties of powers.

This packet goes over the basic information about powers. The BASICALLY AWESOME information about powers, that is!

On a more serious note, though...

The packet also teaches the Power of a Power property, Product of Powers property, and Power of a Product property.

This videos give examples, and the pictures are very clear, colorful, and easy to understand.

Tutorial

A **POWER** is an expression that denotes a number multiplied by itself many times. This sounds complicated, but is really quite easy.

They are written like this:

3^{2}

There are two parts of a power, the **BASE**--

3

and the **EXPONENT**

^{2}

A power signifies that the base is multiplied by itself the number of times in the exponent, so...

3^{2} = 3 x 3

(you multiply 3 two times)

This equals 9

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2^{5} = 2 x 2 x 2 x 2 x 2

(you multiply 2 five times)

This equals 32

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What is 4^{2}?

Powers have 3 important properties:

__Product of Powers__

When a power is multiplied to a power with the same base, the answer has the base raised to the power of the exponents added together.

**a ^{m }x a^{n} = a^{(m+n)}**

Here's why:

4^{2} x 4^{3}

well, 4^{2} = 4 x 4 and 4^{3} = 4 x 4 x 4

put together, it's 4 x 4 x 4 x 4 x 4

or

4^{5 }

[ 2 + 3 = 5 ]

** ****Power of a Power**

When a power is raised to a power, the answer has the base with the two powers multiplied together.

**(a ^{m})^{n = }a^{(mn)}**

Here's why:

(6^{4})^{2}

well, 6^{4} = 6 x 6 x 6 x 6, and since they are squared, it can be represented by:

(6 x 6 x 6 x 6) x (6 x 6 x 6 x 6) OR 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6

Simply

6^{8}

[ 4 x 2 = 8]

** ****Power of a Product**

When the base of a power has multiple terms multiplied together, the exponent gets distributed to all of the terms.

**(ab) ^{m} = a^{m} x b^{m}**

Here's why:

(2Y)^{3}

Well, (2Y)^{3} = (2 x 2 x 2) x (Y x Y x Y)

(2^{3}) x (Y^{3})

Without parentheses:

2^{3 }x Y^{3}

Check out the videos and diagrams below to see these properties in action!

Learn how to use the Power of a Power property. Narrated by Rachel Kaplan.