Exponents

1. Relate and apply the concept of exponents (incl. zero). 2 1. Relate and apply the concept of exponents (incl. zero). 2. Perform calculations following proper order of operations. 3. Applying laws of exponents to compute with integers. 4. Naming square roots of perfect squares through 225.

EXPONENT LAWS

Basic Terminology EXPONENT means BASE

IMPORTANT EXAMPLES

Variable Expressions

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

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

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

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

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

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

ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1.

DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base.

DIVISION PROPERTIES POWER OF A QUOTIENT Hard Example

ZERO, NEGATIVE, AND DIVISION PROPERTIES Zero power Quotient of powers Negative Exponents Power of a quotient

0²=0 6²=36 12²=144 1²=1 7²=49 13²=169 2²=4 8²=64 15²=225 3²=9 9²=81 16²=256 4²=16 10²=100 20²=400 5²=25 11²=121 25²=625

Exponents in Order of Operations 1) Parenthesis →2) Exponents 3) Multiply & Divide 4) Add & Subtract

Exponents & Order of Operations

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

180 – 5 · 2²

Answer: 160

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

Answer: 1

-3²

Answer: -9

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

Contest Problems

Are you ready? 3, 2, 1…lets go!

8(6² - 3(11)) ÷ 8 + 3

Answer: 6

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

Answer: -6

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

Answer: -96

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

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

Answer: 2⁸

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

Answer: 2⁵

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

Answer: 10n⁸

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

Answer: 30r³

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

Answer: 12x³

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

Answer: 36x⁵y⁴

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

Answer: 80x⁶y⁶

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

Answer: 1

(-4)³

Answer: -64

(-2)⁴

Answer: 16

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.

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

(5²) (2⁵) (-1) + 1

Answer: 0

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

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

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

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

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

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

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

Exponent Rule: (a)¯ⁿ = 1÷aⁿ Example: 2¯⁵ = 1 ÷ 2⁵ = 1/32

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

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

Answer: 5³ (125)

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

Answer: 1/2

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

Answer: 3r² 2

Simplify. Your answer should contain only positive exponents. 3xy 5x² 2 ( )

Answer: 9y² 25x²

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

Answer: 9x⁵y⁸ 5

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

Answer: a⁶

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

Answer: 27a⁶

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

Answer: 2⁹

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

Answer: 2⁹

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

Answer: x¹²y¹²

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

Answer: 8x¹²y¹²

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

Answer: 64x²⁴

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

Answer: 16n¹⁰

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

-3 - (1)¯⁵

Answer: -4

(2)¯³

Answer: 1/8

(-2)¯³

Answer: - 1/8

-2⁽¯⁴⁾

Answer: - 1/16

(2)¯³ · (-16)

Answer: -2

56 · (2)¯³

Answer: 7

56 ÷ (2)¯³

Answer: 448

1 ÷ (-3)¯²

Answer: 9

(2²)³ · (6 – 7)² - 2·3² ÷ 6

Answer: 61

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

Answer: -20

2 (10² + 3 · 18) ÷ (5² ÷ 2¯²)

Answer: 3.08

Simplify: (x⁴y¯²)(x¯¹y⁵)

Answer: x³y³

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

Simplify: (4x4y)3 (2xy3)

Answer: 128x13y6

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

Answer: 1

Simplify:

Answer:

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

Answer: -540

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

Answer: -27,000

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

Answer: 6480

Simplify: 24 – 4(12 – 32 – 60)

Answer: 16

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

Answer: 2880

Express in simplest form:

Answer:

Simplify:

Answer: 32

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

Answer: b-3c-3

Simplify. 2 2 3 2 4 2 5 2 59 2 3 4 5 6 … 60 · · · · ·

Answer: 1/900

Solve for n:

Answer: n = 2/3

. Solve for q:

Answer: no solution

. Simplify:

Answer: