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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

This lesson will discuss how powers of monomials are affected when performing basic operations such as multiplication, division, and exponentiation on monomials.

Catch-Up and Review

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

Monomial
Degree of a Monomial

Challenge

Reconsidering Multiplication

Consider the following multiplication expression. lr & 16 * & 9 How can we rewrite this expression to solve it using a multiplication table?

Discussion

Power of a Power Property

A power with a nonzero 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 nonzero 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

Kriz is playing a video game caled Blockfall.

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 increases exponentially.

Suppose Kriz completes 3 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 Simplify 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 We can simplify the expression 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

After clearing three lines of the same color, Kriz's score is x^6y^9.

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!

In a racing game, passing over a green panel increases the speed of the car. However, there are also purple panels that will slow the car 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

First, Kriz's speed is s. After passing the green panel, the speed is multiplied by x. Then, after passing the purple panel, the speed is multiplied by x^(- 1). s ■ → s x ■ → s x x^(- 1) Let's simplify this monomial. We start by using the Product of Powers Property. We also use the fact that if a number or variable has no exponent written, its exponent is 1.

sxx^(- 1)

a=a^1

sx^1 x^(- 1)

MultPow

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

sx^(1 + (- 1))

AddNeg

a+(- b)=a-b

sx^(1-1)

SubTerms

Subtract terms

sx^0

Next, the Zero Exponent Property tells us that x^0 = 1, and any number or variable multiplied by 1 is equal to itself.

sx^0

ExponentZero

a^0=1

s* 1

MultByOne

a * 1=a

Kriz's speed after passing over the green and purple panels is simply s.

Example

The Hardest Level

In a video game, a player starts with a score of 1. The score is multiplied by x each time an enemy is defeated and divided by x each time the player loses a life.

A player defeats 5 enemies and loses 7 lives. 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

The player defeats a total of 5 enemies. This gives them a score of x raised to the power of 5. x^5 Unfortunately, they also lose 7 times. This means their score will be divided by x raised to the power of 7. x^5/x^7 We can simplify the expression 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)

Since none of the options in the pool has a negative exponent, let's use the Negative Exponent Property to rewrite it with a positive exponent.

x^(- 2)

NegExponentToFrac

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

1/x^2

The player's score is 1x^2.

Closure

Solving the Problem With a Table

Recall the multiplication expression from the beginning of the lesson. lr & 16 * & 9 Both 16 and 9 are perfect squares. We can find these values on the multiplication table.

Let's 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, we 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

We can also use the multiplication table to find the product of 4 and 3.

This means the above product can be written as 12^2.

(4 * 3)^2

Multiply

12^2

This value is also on the table.

Now we can solve the multiplication expression.

lr & 16 * & 9 & 144

Level 1

Level 2

Level 3

Powers of Monomials

Exercises

Raise the monomials to the indicated power.

(x^2)^3

(y^3)^2

(z^4)^5

Let's take a look at the first given expression. (x^2)^3 We can use the Power of a Power Property to write the expression as a single monomial raised to a power. (x^2)^3 &= x^(2 * 3) &= x^6 This means that the answer is x^6.

This time we are given the following expression. (y^3)^2 We can rewrite this expression using the Power of a Power Property as well. (y^3)^2 &= y^(3 * 2) &= y^6 The answer is y^6.

Let's take a look at the last expression. (z^4)^5 We will use the Power of a Power Property one more time. (z^4)^5 &= z^(4 * 5) &= z^(20) The answer is z^(20).

Divide the monomials. Write the answer using only positive exponents.

x^6/x^6

y^2/y^7

z^5/z^9

Let's begin by taking a look at the given expression. x^6/x^6 The above expression involves a quotient, so we will use the Quotient of Powers Property.

We are left with x to the power of 0. The Zero Exponent Property tells us that any number or variable raised to the power of zero is equal to 1. x^0 = 1 This means that the answer is 1. Note that we can also reach the same conclusion by noting that we are dividing x^6 by itself. x^6/x^6 = 1

Let's take a look at the next expression. y^2/y^7 Use the Quotient of Powers Property and subtract the power of the denominator from the power of the numerator.

We are asked to write our answer using only positive exponents. This means that we should use the Negative Exponent Property to rewrite our answer.

Finally, let's take a look at the last expression. z^5/z^9 Use the Quotient of Powers Property once again.

Multiply the expressions. Write the answer using only positive exponents.

a^3 * a^(- 7)

b^5 * b^(- 5)

c^(- 4) * c^2

Let's begin by looking at the given expression. a^3 * a^(- 7) The expression is a product that involves negative exponents. Let's use the Product of Powers Property to multiply them.

We are asked to write our answer using only positive exponents. This means that we should use the Negative Exponent Property to rewrite our answer.

The next expression is also a product that involves negative exponents. b^5 * b^(- 5) Use the Product of Powers Property to multiply the above expressions.

We are left with b to the power of 0. The Zero Exponent Property tells us that any number or variable raised to the power of zero is equal to 1. b^0 = 1 This means that the answer is 1.

Let's take a look at the last expression. c^(- 4) * c^2 Once again, use the Product of Powers Property. Finish by using the Negative Exponent Property to write the negative exponent as positive.

Multiply the expressions. Write the answer using only positive exponents.

1/a^2 * 1/a^4

1/b * 1/b^6

1/c^7 * c^2

Consider the given expression. 1/a^2 * 1/a^4 To find the product, let's begin by rewriting the quotients using the Negative Exponent Property.

Next, we will find the product using the Product of Powers Property.

We are asked to give our answer using positive exponents. This means we should use the Negative Exponent Property once again.

Let's take a look at the next expression. 1/b * 1/b^6 Rewrite each factor using the Negative Exponent Property.

We can now multiply using the Product of Powers Property.

Finish by writing the negative exponent as a positive using the Negative Exponent Property.

Consider the last expression. 1/c^7 * c^2 Rewrite the first factor using the Negative Exponent Property.

Then, use the Product of Powers Property.

Rewrite the final answer using the Negative Exponent Property.

A student tried to do an arithmetical operation. However, they got a wrong answer.

In which step did the student make the mistake?

Which of the following is the correct answer?

A student was trying to do the following arithmetical operation. 3^(- 2) The above expression has a negative exponent, so use the Negative Exponent Property to rewrite it. 3^(- 2) = 1/3^2 This is where the student made their mistake. They used the Negative Exponent Property incorrectly. This means that the answer is Step I.

We can find the correct answer by continuing with the operation in the correct way.

The correct answer is 19.

Powers of Monomials

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