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Exponents

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**1. Relate and apply the concept of exponents (incl. zero). 2**

1. Relate and apply the concept of exponents (incl. zero) Perform calculations following proper order of operations Applying laws of exponents to compute with integers Naming square roots of perfect squares through 225.

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EXPONENT LAWS

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Basic Terminology EXPONENT means BASE

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IMPORTANT EXAMPLES

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Variable Expressions

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**Substitution and Evaluating**

STEPS Write out the original problem. Show the substitution with parentheses. Work out the problem. = 64

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**Evaluate the variable expression when x = 1, y = 2, and w = -3**

Step 1 Step 1 Step 1 Step 2 Step 2 Step 2 Step 3 Step 3 Step 3

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**MULTIPLICATION PROPERTIES**

PRODUCT OF POWERS This property is used to combine 2 or more exponential expressions with the SAME base.

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**MULTIPLICATION PROPERTIES**

POWER TO A POWER This property is used to write and exponential expression as a single power of the base.

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**MULTIPLICATION PROPERTIES**

POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions.

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**MULTIPLICATION PROPERTIES**

SUMMARY PRODUCT OF POWERS ADD THE EXPONENTS POWER TO A POWER MULTIPLY THE EXPONENTS POWER OF PRODUCT

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**ZERO AND NEGATIVE EXPONENTS**

ANYTHING TO THE ZERO POWER IS 1.

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**DIVISION PROPERTIES QUOTIENT OF POWERS**

This property is used when dividing two or more exponential expressions with the same base.

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DIVISION PROPERTIES POWER OF A QUOTIENT Hard Example

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**ZERO, NEGATIVE, AND DIVISION PROPERTIES**

Zero power Quotient of powers Negative Exponents Power of a quotient

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0²= ²= ²=144 1²= ²= ²=169 2²= ²= ²=225 3²= ²= ²=256 4²= ²= ²=400 5²= ²= ²=625

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**Exponents in Order of Operations**

1) Parenthesis →2) Exponents 3) Multiply & Divide 4) Add & Subtract

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**Exponents & Order of Operations**

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**Contest Problems Are you ready? 3, 2, 1…lets go!**

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180 – 5 · 2²

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Answer: 160

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**Evaluate the expression when y= -3 (2y + 5)²**

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Answer: 1

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-3²

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Answer: -9

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**Warning. The missing parenthesis makes all the difference**

Warning!!! The missing parenthesis makes all the difference. The square of a negative & the negative of a square are not the same thing! Example: (-2)² ≠ -2²

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Contest Problems

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Are you ready? 3, 2, 1…lets go!

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8(6² - 3(11)) ÷ 8 + 3

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Answer: 6

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**Evaluate the expression when a= -2 a² + 2a - 6**

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Answer: -6

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**Evaluate the expression when x= -4 and t=2 x²(x-t)**

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Answer: -96

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**Exponent Rule: a ∙ aⁿ = a Example1: 2 ∙ 2 = 2¹⁺¹ = 2² = 4**

m + n Example1: 2 ∙ 2 = 2¹⁺¹ = 2² = 4 Example2: 2³ ∙ 2² = 2³⁺² = 2⁵ = 32

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**Simplify (in terms of 2 to some power)**

Simplify (in terms of 2 to some power). Your answer should contain only positive exponents. 4² · 4²

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Answer: 2⁸

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**Simplify (in terms of 2 to some power)**

Simplify (in terms of 2 to some power). Your answer should contain only positive exponents · 2² · 2²

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Answer: 2⁵

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**Simplify. Your answer should contain only positive exponents**

Simplify. Your answer should contain only positive exponents. 2n⁴ · 5n ⁴

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Answer: 10n⁸

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**Simplify. Your answer should contain only positive exponents. 6r · 5r²**

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Answer: 30r³

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**Simplify. Your answer should contain only positive exponents. 6x · 2x²**

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Answer: 12x³

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**Simplify. Your answer should contain only positive exponents**

Simplify. Your answer should contain only positive exponents. 6x² · 6x³y⁴

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Answer: 36x⁵y⁴

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**Simplify. Your answer should contain only positive exponents**

Simplify. Your answer should contain only positive exponents. 10xy³ · 8x⁵y³

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Answer: 80x⁶y⁶

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**Simplify Completely. Your answer should not contain exponents. 3⁵ · 3¯⁵**

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Answer: 1

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(-4)³

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Answer: -64

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(-2)⁴

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Answer: 16

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Important! *If a negative number is raised to an even number power, the answer is positive. *If a negative number is raised to an odd number power, the answer is negative.

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**Contest Problem Are you ready? 3, 2, 1…lets go!**

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(5²) (2⁵) (-1)

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Answer: 0

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**Exponent Rule: (ab)² = a²b² Example: (4·6)² = 4²·6²**

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**Exponent Rule: (a/b)² = a²/b² Example: (7/12)² = 7²/12² = 49/144**

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**Exponent Rule: (a÷b)ⁿ = aⁿ÷bⁿ = aⁿ/bⁿ**

Example: (2÷5)³ = (2÷5)·(2÷5)·(2÷5) = (―)·(―)·(―) =(2·2·2)/(5·5·5) =2³/5³ = 8/125 2 5 2 5 2 5

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**Exponent Rule: (1/a)² = 1/a² Example: (1/7)² = 1/7² = 1/49**

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Exponent Rule: a ÷aⁿ = a m m - n Example: 2⁵ ÷ 2² = 2⁵¯² = 2³ = 8

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m · n m Exponent Rule: (a )ⁿ = a 2·5 Example: (2²)⁵ = 2 = 2¹⁰ = 1,024

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Exponent Rule: a⁰ = 1 Examples: (17)⁰ = 1 (99)⁰ = 1

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**Exponent Rule: (a)¯ⁿ = 1÷aⁿ**

Example: 2¯⁵ = 1 ÷ 2⁵ = 1/32

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**Problems Are you ready? 3, 2, 1…lets go!**

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**Simplify. Your answer should contain only positive exponents. 5⁴ 5**

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Answer: 5³ (125)

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**Simplify. Your answer should contain only positive exponents. 2² 2³**

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Answer: 1/2

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**Simplify. Your answer should contain only positive exponents. 3r³ 2r**

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Answer: 3r² 2

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**Simplify. Your answer should contain only positive exponents. 3xy 5x²**

2 ( )

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Answer: 9y² 25x²

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**Simplify. Your answer should contain only positive exponents**

Simplify. Your answer should contain only positive exponents. 18x⁸y⁸ 10x³

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Answer: 9x⁵y⁸ 5

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**Simplify. Your answer should contain only positive exponents. (a²)³**

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Answer: a⁶

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**Simplify. Your answer should contain only positive exponents. (3a²)³**

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Answer: 27a⁶

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**Simplify. Your answer should contain only positive exponents. (2³)³**

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Answer: 2⁹

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**Simplify. Your answer should contain only positive exponents. (8)³**

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Answer: 2⁹

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**Simplify. Your answer should contain only positive exponents. (x⁴y⁴)³**

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Answer: x¹²y¹²

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**Simplify. Your answer should contain only positive exponents. (2x⁴y⁴)³**

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Answer: 8x¹²y¹²

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**Simplify. Your answer should contain only positive exponents. (4x⁴∙x⁴)³**

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Answer: 64x²⁴

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**Simplify. Your answer should contain only positive exponents. (4n⁴∙n)²**

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Answer: 16n¹⁰

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**Simplify the following problems completely**

Simplify the following problems completely. Your answer should not contain exponents. Example: 2³·2² = 2⁵ = 32

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-3 - (1)¯⁵

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Answer: -4

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(2)¯³

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Answer: 1/8

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(-2)¯³

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Answer: - 1/8

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-2⁽¯⁴⁾

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Answer: - 1/16

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(2)¯³ · (-16)

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Answer: -2

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56 · (2)¯³

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Answer: 7

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56 ÷ (2)¯³

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Answer: 448

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1 ÷ (-3)¯²

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Answer: 9

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(2²)³ · (6 – 7)² - 2·3² ÷ 6

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Answer: 61

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-6 - (-4)(-5) - (-6)

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Answer: -20

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2 (10² + 3 · 18) ÷ (5² ÷ 2¯²)

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Answer: 3.08

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**Simplify: (x⁴y¯²)(x¯¹y⁵)**

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Answer: x³y³

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Competition Problems Points: 1 minute: 5 points 1 ½ minute: 3 points 2 minute: 1 point 3, 2, 1, … go!

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Simplify: (4x4y)3 (2xy3)

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Answer: 128x13y6

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**If A = (7 – 11 + 8)131 and B = (–7 + 11 – 8)131 then what is the value of: (7 – 13)(A+B)**

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Answer: 1

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Simplify:

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Answer:

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**Evaluate for x = –2, y = 3 and z = –4:**

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Answer: -540

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**If A♣B = (3A–B)3, then what is (2♣8)♣6?**

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Answer: -27,000

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**If a*b is defined as (ab)2 + 2b, and x y is defined as xy2 - 2y, find 2*(3 4).**

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Answer: 6480

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Simplify: 24 – 4(12 – 32 – 60)

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Answer: 16

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**If x = the GCF of 16, 20, and 72 and y = the LCM of 16, 20, and 72, what is xy?**

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Answer: 2880

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**Express in simplest form:**

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Answer:

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Simplify:

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Answer: 32

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**Simplify. Write the answer with negative exponents. (abc)-3c2b a-4bc2a**

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Answer: b-3c-3

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Simplify …

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Answer: 1/900

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Solve for n:

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Answer: n = 2/3

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. Solve for q:

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Answer: no solution

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. Simplify:

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Answer:

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