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It is possible to perform basic operations such as multiplication, division, and exponentiation on monomials. This lesson will discuss how powers of monomials operate.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

## Finish Your Homework Before Playing!

Kriz wants to play some video games with their little cousin. However, Kriz's cousin has not finished his homework yet.

Kriz does not want to spill out the answer right away. They want to help their cousin understand math! Kriz's cousin only knows how to use multiplication tables.

Help Kriz rewrite the multiplication in a way that it can be solved using the multiplication table.
Discussion

## Power of a Power Property

A power with a non-zero base and an integer exponent that is raised to another integer exponent can be written as a power with base and exponent

For the rule to be true for both exponents must be greater than zero.
Discussion

## Power of a Product Property

A power with an integer exponent whose base is the product of two non-zero factors and can be written as the product of two powers with bases and and the same exponent

For this rule to be valid when either or is must be greater than zero.
Example

## Bonus for Same Color

While waiting for their cousin to finish his homework,Kriz plays a single-player game video game.

This game might look familiar! Kriz is particularly good at this video game, so they decide to try a new challenge level. In this level, Kriz's score is represented by a monomial.
Each time Kriz completes a line of the same color, their score is raised by a power.
Suppose Kriz completes three lines of the same color. Their score will be given by raising the monomial to the third power.
Write the monomial that represents Kriz's score.

### Solution

After dropping the green L-shaped block, Kriz will complete three lines of the same color.
This means that their score will be raised to the third power.
The expression can be expanded by using the Power of a Product Property and the Power of a Power Property.
Kriz's score becomes after clearing three lines of the same color. What a skilled player!
Discussion

## Zero Exponent Property

Any non-zero real number raised to the power of is equal to

Discussion

## Negative Exponent Property

If is a non-zero real number and is a positive integer, then raised to the power of is equal to over raised to the power of

It is important to note that the sign of the exponent changes, meaning that the resulting fraction is not simply the reciprocal of the original power.
Example

## Watch the Boosters!

Still waiting for their cousin, Kriz switches to a racing game. In this game, passing through a green panel increases the speed of the kart. However, there are also purple panels that will slow the kart down.
Kriz's initial speed is represented by Hitting a green panel multiplies this speed by Hitting a purple panel multiplies the speed by
On the first lap, Kriz crosses a green panel, followed by a purple panel. Which of the following expressions represents Kriz's speed after passing over the panels?

### Hint

Use the Product of Powers Property and the Zero Exponent Property. Remember, any number or expression multiplied by is equal to itself.

### Solution

After passing over a green panel, Kriz gets a boost to their speed represented by This bonus speed multiplies the current speed of the kart
Right after that, they hit a purple panel. This will slow the kart down by multiplying its speed by
If a number or a variable has no power written on it, it is assumed to be
This monomial can be simplified using the Product of Powers Property.
Because of the Zero Exponent Property, is equal to Any number or variable multiplied by is equal to itself. This means that the speed of Kriz's kart after passing over the green and purple panels is simply
Example

## The Cousins Play the Hardest Level

Kriz's cousin has finally finished his homework, so they can play a cooperative game together.

Their score is multiplied by for each enemy they slay. However, their score is also divided by every time either player is taken down.
Kriz accidentally starts the hardest level. They slay enemies together, but they also get taken down times. Which of the following expressions represents their final score?

### Hint

Use the Quotient of Powers Property and the Negative Exponent Property to reach the final answer.

### Solution

While playing in the hardest level, Kriz and their cousin slay a total of enemies. This will give them a score of raised to the power of
Unfortunately, they also get taken down a total of times. This is the hardest level, after all! This means that their score will be divided by raised to the power of
The expression can be simplified using the Quotient of Powers Property.
Their score is represented by Since no option from the pool is written with a negative exponent, use the Negative Exponent Property to rewrite it with a positive exponent. The previous expression represents Kriz and his cousin's score.
Closure

## Solving the Problem With a Table

Kriz's cousin needs to finish his homework. He is down to the last multiplication problem.

Note that both and are perfect squares. This can be shown in the multiplication table.

Use this information to rewrite the original multiplication.
Next, use the Power of a Product Property to rewrite the product. Find the product of and in the table.
This means the above product can be written as squared.
Twelve squared is also on the multiplication table.

Finally, the answer can be given using only information from the multiplication table.