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Pre-Algebra View details
9. Powers of Monomials
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Chapter 4
9. 

Powers of Monomials

Navigating the world of monomials becomes easier when you understand the power property, zero exponent, and negative exponent. This lesson explains how these properties work and why they are crucial in algebra. For instance, the power property allows to simplify complex expressions, making calculations more manageable. Understanding the zero exponent rule can help you quickly solve equations, while the negative exponent concept is vital for dealing with fractions in algebraic expressions. These foundational skills are not just academic exercises; they are practical tools that you'll use in various fields, from engineering to finance.
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Lesson Settings & Tools
10 Theory slides
9 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Powers of Monomials
Slide of 10
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.

Sixteen times nine equals?

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.

A multiplication table showing products of values from 1 to 12 in columns and rows.
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.

Color Blocks Videogame
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.
Color Blocks Videogame
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.

Hint

Use the Power of a Product Property and the Power of a Power Property.

Solution
After dropping the green L-shaped block, Kriz will complete three lines of the same color.
Color Blocks Videogame
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.
PowProd

PowPow

Multiply

Multiply

Multiply

Multiply

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.
MultPow

AddNeg

SubTerm

Subtract term

Because of the Zero Exponent Property, is equal to
ExponentZero

Any number or variable multiplied by is equal to itself.
MultByOne

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.

Video game cover
Their score is multiplied by for each enemy they slay. However, their score is also divided by every time either player is taken down.
Video Game
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.
DivPow

SubTerm

Subtract term

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.
NegExponentToFrac

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.

Sixteen times nine equals?

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

Multiplication table with 4 times 4 and 3 times 3 highlighted
Use this information to rewrite the original multiplication.
Rewrite

Rewrite as

Rewrite

Rewrite as

Next, use the Power of a Product Property to rewrite the product.
ProdPow

Find the product of and in the table.
Multiplication table with the product of three and four highlighted
This means the above product can be written as squared.
Multiply

Multiply

Twelve squared is also on the multiplication table.
Multiplication table with the product of twelve and twelve highlighted

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


Level 1
Level 2
Level 3
Powers of Monomials
Pair the numbers that are equal in value.

We are asked to pair the numbers that are equal in value. Let's start by noting that every number of the left column is written as a power of either 2 or 3. 2^4 2^6 3^4 3^8 Let's try to write the bases of the numbers in the right column as powers of 2 and of 3 as well. This will make it easier to match the two columns. 4 = 2^2 8 = 2^3 9 = 3^2 Now we can use the Power of a Power Property on each of the possible values on the right to simplify them. Let's rewrite 4^2 first. 4^2 &= (2^2)^2 &= 2^(2 * 2) &= 2^4 We can rewrite the rest of the numbers in a similar fashion.

Number Rewrite Use Power of a Power Property Simplify
4^2 (2^2)^2 2^(2 * 2) 2^4
8^2 (2^3)^2 2^(3 * 2) 2^6
9^2 (3^2)^2 3^(2 * 2) 3^4
9^4 (3^2)^4 3^(2 * 4) 3^8

We can now pair every number. 2^4 → 4^2 2^6 → 8^2 3^4 → 9^2 3^8 → 9^4

Sort the following numbers from least to greatest.

We are asked to sort the given list of numbers from least to greatest. Let's start with 2^0. We know that because of the Zero Exponent Property, 2^0 is equal to 1. 2^0 = 1 Next, note that 2^3 means that 2 is being multiplied by itself three times. Likewise, 2^6 means that 2 is being multiplied by itself six times. This means that 2^6 is greater than 2^3. We can use this information to sort the numbers with non-negative powers. 2^0 < 2^3 < 2^6 Let's now use the Negative Exponent Property to convert the negative powers into positive powers. 2^(- 1) &= 1/2 [0.8em] 2^(- 3) &= 1/2^3 Both of these numbers are less than 1, but we still need to determine which is less than the other. Note that 2^3 is equal to 8, so let's rewrite 2^(- 3). 2^(-3) &= 1/2^3 [0.8em] &= 1/8 This means that 2^(- 1) is one half and 2^(- 3) is one-eighth, and we know that 18 is less than 12.

Knowing this, we can sort all of the numbers from least to greatest. &2^(- 3) &2^(- 1) &2^0 &2^3 &2^6

Consider a cube with a side length of

Cube with side length 1/x
Which of the following expressions corresponds to the volume of the cube?

We want to find the volume of the given cube.

Let's use the formula for the volume of a cube. V=s^3 Substitute 1x for s in the above formula. V= (1/x)^3 Next, use the Negative Exponent Property to rewrite the right-hand side of the above equation. V &= ( 1/x ) ^3 [0.8em] &= (x^(- 1))^3 We can now use the Power of a Power Property to simplify the expression. V &= (x^(- 1))^3 [0.75em] &= x^(- 1 * 3) [0.75em] &= x^(- 3) This means that the volume of the cube is given by the expression x^(- 3).

Powers of Monomials

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