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Pre-Algebra View details

9. Powers of Monomials

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Pre-Algebra /

Chapter 4

Powers of Monomials

Navigating the world of monomials becomes easier when we understand the Power Property and the properties of zero exponents and negative exponents. This lesson explains how these properties work and why they are crucial in algebra. For instance, the Power Property allows us to simplify complex expressions, making calculations more manageable. Understanding the Zero Exponent Rule can help us 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 are used in various fields, from engineering to finance.

Student Learning Objectives:
Use the Power of a Power, Power of a Product, Zero Exponent, and Negative Exponent Properties to simplify monomials and algebraic expressions

	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.

Monomial
Degree of a Monomial

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 a and an integer exponent m that is raised to another integer exponent n can be written as a power with base a and exponent m* n.

(a^m)^n = a^(m* n)

For the rule to be true for a=0, both exponents must be greater than zero.

Discussion

Power of a Product Property

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

(ab)^m = a^m b^m

For this rule to be valid when either a or b is 0, m 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. x^2y^3 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 x^2y^3 to the third power. (x^2y^3)^3 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.

This means that their score will be raised to the third power. (x^2y^3)^3 The expression can be expanded by using the Power of a Product Property and the Power of a Power Property.

(x^2y^3 )^3

PowProd

(a * b)^m=a^m* b^m

( x^2 ) ^3 * ( y^3)^3

PowPow

(a^m)^n=a^(m* n)

(x^(2 * 3)) * ( y^(3 * 3))

Multiply

x^6 * y^9

Multiply

x^6y^9

Kriz's score becomes x^6y^9 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 0 is equal to 1.

a^0=1

Discussion

Negative Exponent Property

If a is a non-zero real number and n is a positive integer, then a raised to the power of - n is equal to 1 over a raised to the power of n.

a^(- n)=1/a^n

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 s. Hitting a green panel multiplies this speed by x. Hitting a purple panel multiplies the speed by x^(- 1).

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 1 is equal to itself.

Solution

After passing over a green panel, Kriz gets a boost to their speed represented by x. This bonus speed multiplies the current speed of the kart s. s * x = sx Right after that, they hit a purple panel. This will slow the kart down by multiplying its speed by x^(- 1). sx * x^(- 1) = sxx^(- 1) If a number or a variable has no power written on it, it is assumed to be 1. sxx^(- 1) = sx^1x^(- 1) This monomial can be simplified using the Product of Powers Property.

sx^1x^(- 1)

MultPow

a^m*a^n=a^(m+n)

sx^(1+(-1))

AddNeg

a+(- b)=a-b

sx^(1-1)

SubTerm

Subtract term

sx^0

Because of the Zero Exponent Property, x^0 is equal to 1.

sx^0

ExponentZero

a^0=1

s * 1

Any number or variable multiplied by 1 is equal to itself.

s * 1

MultByOne

a * 1=a

This means that the speed of Kriz's kart after passing over the green and purple panels is simply s.

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 x for each enemy they slay. However, their score is also divided by x every time either player is taken down.

Kriz accidentally starts the hardest level. They slay 5 enemies together, but they also get taken down 7 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 5 enemies. This will give them a score of x raised to the power of 5. x^5 Unfortunately, they also get taken down a total of 7 times. This is the hardest level, after all! This means that their score will be divided by x raised to the power of 7. x^5/x^7 The expression can be simplified using the Quotient of Powers Property.

x^5/x^7

DivPow

a^m/a^n= a^(m-n)

x^(5-7)

SubTerm

Subtract term

x^(- 2)

Their score is represented by x^(- 2). Since no option from the pool is written with a negative exponent, use the Negative Exponent Property to rewrite it with a positive exponent.

x^(- 2)

NegExponentToFrac

a^(- m)=1/a^m

1/x^2

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 16 and 9 are perfect squares. This can be shown in the multiplication table.

Use this information to rewrite the original multiplication.

16 * 9

Rewrite

Rewrite 16 as 4^2

(4^2) * 9

Rewrite

Rewrite 9 as 3^2

(4^2) * (3^2)

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

(4^2) * (3^2)

ProdPow

a^m* b^m=(a * b)^m

(4 * 3)^2

Find the product of 3 and 4 in the table.

This means the above product can be written as 12 squared.

(4* 3)^2

Multiply

12^2

Twelve squared is also on the multiplication table.

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

16 * 9 = 144

Level 1

Level 2

Level 3

Powers of Monomials

Exercise 2.1

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 1x.

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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