Exponents
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 BASE EXPONENT means
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
Write the following expression using an exponent: 2·2·2·2·2·2
Answer: 2⁶
Simplify the following problems completely. Your answer should not contain exponents. 3³
Answer: 27
Simplify the following problems completely. Your answer should not contain exponents. Example: 2³·2² = 2⁵ = 32 Are you ready? 3, 2, 1…lets go!
(-4)³
Answer: -64
(-2)³
Answer: -8
(-2)⁴
Answer: 16
(-2)⁵
Answer: -32
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.
(-2)²
Answer: 4
(-12)²
Answer: 144
-3²
Answer: -9
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
IMPORTANT EXAMPLES
Exponent Rule: a ∙ aⁿ = a m + nm Example2: 2³ ∙ 2² = 2³⁺² = 2⁵ = 32 Example1: 2 ∙ 2 = 2¹⁺¹ = 2² = 4
Are you ready? 3, 2, 1…lets go!
Simplify. Your answer should contain only positive exponents. 2 · 2² · 2²
Answer: 2⁵
Simplify. Your answer should contain only positive exponents. 4² · 4²
Answer: 4⁴
Simplify. Your answer should contain only positive exponents. 4 · 4²
Answer: 4³
Simplify. Your answer should contain only positive exponents. 3² · 3²
Answer: 3⁴
Simplify Completely. Your answer should not contain exponents. 3⁵ · 3¯⁵
Answer: 1
(-1) + 1 (5²) (2⁵) Simplify Completely. Your answer should not contain exponents.
Answer: 0
Exponent Rule: (a )ⁿ = a Example: ( 2² )⁵ = 2 = 2¹⁰ = 1,024 m m · n 2·5
Contest Problem Are you ready? 3, 2, 1…lets go!
Simplify. Your answer should contain only positive exponents. (2³)³
Answer: 2⁹
Simplify. Express your answer with a base of 2 and an exponent. Example: 2² (8)³
Answer: 2⁹
Simplify. Express your answer with a base of 2 and an exponent. Example: 2² 4² · 4²
Answer: 2⁸
Simplify. Express your answer with a base of 2 and an exponent. Example: 2² 4 · 4²
Answer: 2⁶
Exponent Rule: (ab)² = a²b² Example: (4·6)² = 4²·6²
Simplify Completely. Your answer should not contain exponents. 5³ · 2³
Answer: 1000
Simplify Completely. Your answer should not contain exponents. (5·4)³
Answer: 8000
Exponent Rule: a ÷aⁿ = a m - nm Example: 2⁵ ÷ 2² = 2⁵¯² = 2³ = 8
Problems Are you ready? 3, 2, 1…lets go!
Evaluate Completely: 5⁴ 5
Answer: 125
Evaluate Completely: 2² 2³
Answer: 1/2
Exponent Rule: (1/a)² = 1/a² Example: (1/7)² = 1/7² = 1/49
Simplify. (1/4)³
Answer: 1/64
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/
Problems Are you ready? 3, 2, 1…lets go!
Simplify. (2/3)³
Answer: 8/27
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!
(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
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!
-6 - (-4)(-5) - (-6)
Answer: -20
180 – 5 · 2²
Answer: 160
-3 - (1)¯⁵
Answer: -4
8 ( 6² - 3(11) ) ÷ 8 + 3
Answer: 6
( 2² )³ · (6 – 7)² - 2·3² ÷ 6
Answer: 61
2 ( 10² + 3 · 18 ) ÷ ( 5² ÷ 2¯² )
Answer: 3.08