Vocabulary

Base Exponent

Scientific Notation

Objective 1

You will be able to simplify expressions with numbers and variables using properties of exponents

Warm-Up, 1

Consider the algebraic expression . We usually think of this as distributed through the parenthesis, but it also means copies of :

2 𝑥+𝑦 +¿ +¿2 𝑥+𝑦 2 𝑥+𝑦

Multiplication is simply repeated addition

Warm-Up, 2

Now consider the algebraic expression . What could be done to the illustration below to represent this new expression?

(2 𝑥+𝑦 ) × ×(2 𝑥+𝑦 ) (2 𝑥+𝑦 )

An exponent is simply repeated multiplication

Exponents

23 = 222Base

Exponent

Exponents mean

repeated multiplication

Exercise 1

1. Write in expanded form

2. Write in expanded form

3. Simplify

4. Simplify

Investigation 1

In this Investigation, we will (re)discover some general properties of exponents. They include the Multiplication and Division Properties, and Power Properties.

Investigation 1: MultiplicationStep 1: Rewrite each product in expanded

form, and then rewrite it in exponential form with a single base.

Step 2: Compare your answers to the original product. Is there a shortcut?

Step 3: Generalize your observations by filling in the blank: bm·bn = b-?-

34·32 103·106 x3·x5 a2·a4

Investigation 1: Powers

Step 1: Rewrite each expression without parentheses.

Step 2: Generalize your observations by filling in the blanks:

(bm)n = b-?-

(ab)n = a-?-b-?-

(45)2 (x3)4 (5m)n (xy)3

Investigation 1: Division

Step 1: Write the numerator and denominator in expanded form, and then reduce to eliminate common factors. Rewrite the factors that remain with exponents.

Step 2: Generalize your observations by filling in the blank:

𝑏𝑚

𝑏𝑛 =𝑏−? −

Properties of Exponents

Multiplication Property of Exponents

𝑏𝒎 ∙𝑏𝒏=𝑏𝒎+𝒏

Division Property of Exponents

𝑏𝒎

𝑏𝒏 =𝑏𝒎−𝒏

Power Property of Exponents

Exercise 2

Practice simplifying expressions.

1. 2.

3. 4.

Exercise 3

Simplify

Not the Power Property

Notice that when expanding , you don’t get to use the Power Property of exponents to “distribute” the exponent through the parenthesis.

The Power Property of Exponents only works across multiplication and division

NOT addition or subtraction!

Exercise 4

Evaluate the expression.1. 2.

3. 4.

Exercise 5

Use the division property of exponents to rewrite each expression with a single exponent. Then expand each original expression and simplify. Compare your answers.

Properties of Exponents

Zero Exponents

𝑏0=1

Negative Exponents

𝑏−𝒏=1

𝑏𝒏1

𝑏−𝒏=𝑏𝒏

Exercise 6

Simplify the expression.1. 2.

3. 4.

Always Look on the Bright Side of Life…

When you simplify an algebraic expression involving exponents, all the exponents must be POSITIVE.

Negative exponents in the numerator need to go in the denominator

𝑎𝑏𝑐−𝑛

𝑑=¿

𝑎𝑏𝑑𝑐𝑛

Always Look on the Bright Side of Life…

When you simplify an algebraic expression involving exponents, all the exponents must be POSITIVE.

Negative exponents in the denominator need to go in the numerator

𝑎𝑏𝑑𝑐−𝑛=¿𝑎𝑏𝑐

𝑛

𝑑

Exercise 7a

Simplify the expression.1. 2.

3. 4.

Exercise 7b

Simplify the expression

Exercise 8The radius of Jupiter is about 11 times greater

than the radius of earth. How many times as great as Earth’s volume is Jupiter’s volume?

𝑉=43𝜋 𝑟3

Exercise 9

The area of a rectangle is units2. Find the length of the rectangle if its width is units.

Exercise 10

Let’s say the number is in scientific notation. What must be true about ? What must be true about ?

Scientific Notation

The number is in scientific notation when and is an integer.

Easy to multiply, divide, and raise to powers

using the properties of exponents

NOT so easy to add and subtract

Exercise 11

Write the answer in scientific notation.

1. 2.

5.1: Use Properties of Exponents

Objectives:

1. To simplify numeric and algebraic expressions using the properties of exponents

Insert your face here