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Vocabulary
Base Exponent
Scientific Notation
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Objective 1
You will be able to simplify expressions with numbers and variables using properties of exponents
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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
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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
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Exponents
23 = 222Base
Exponent
Exponents mean
repeated multiplication
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Exercise 1
1. Write in expanded form
2. Write in expanded form
3. Simplify
4. Simplify
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Investigation 1
In this Investigation, we will (re)discover some general properties of exponents. They include the Multiplication and Division Properties, and Power Properties.
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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
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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
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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−? −
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Properties of Exponents
Multiplication Property of Exponents
𝑏𝒎 ∙𝑏𝒏=𝑏𝒎+𝒏
Division Property of Exponents
𝑏𝒎
𝑏𝒏 =𝑏𝒎−𝒏
Power Property of Exponents
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Exercise 2
Practice simplifying expressions.
1. 2.
3. 4.
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Exercise 3
Simplify
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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!
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Exercise 4
Evaluate the expression.1. 2.
3. 4.
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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.
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Properties of Exponents
Zero Exponents
𝑏0=1
Negative Exponents
𝑏−𝒏=1
𝑏𝒏1
𝑏−𝒏=𝑏𝒏
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Exercise 6
Simplify the expression.1. 2.
3. 4.
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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
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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
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Exercise 7a
Simplify the expression.1. 2.
3. 4.
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Exercise 7b
Simplify the expression
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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
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Exercise 9
The area of a rectangle is units2. Find the length of the rectangle if its width is units.
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Exercise 10
Let’s say the number is in scientific notation. What must be true about ? What must be true about ?
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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
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Exercise 11
Write the answer in scientific notation.
1. 2.
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5.1: Use Properties of Exponents
Objectives:
1. To simplify numeric and algebraic expressions using the properties of exponents
Insert your face here