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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 expand 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

Kriz's score becomes x^6y^9 after clearing three lines of the same color.

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

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) Let's simplify this monomial 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 Kriz's speed after passing over the green and purple panels is simply s.

Example

The Hardest Level

In a video game, a player's 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 that 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)

Their score is represented by x^(- 2). Since no option from the pool is written with 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

This expression represents the player's score.

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, 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

Let's find the product of 3 and 4 in the table.

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

Exercise 3.1

Consider a cube with a side length of xy^5z^3.

Write a monomial that represents 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 xy^5z^3 for s in the formula. V= (xy^5z^3)^3 Next, let's use the Power of a Product Property to expand the monomial. V &= ( xy^5z^3 ) ^3 [0.75em] &= (x)^3 * (y^5)^3 * (z^3)^3 We can now use the Power of a Power Property on each factor to simplify them.

This means that the volume of the cube is given by the expression x^3y^(15)z^9.

Powers of Monomials

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