Working with Powers Definition of Powers Integer Exponents

Working with Powers • Definition of Powers • Integer Exponents • The seven Exponent Rules 1

Repeated multiplications… • 2× 2= • 2× 2× 2× 2= • How can we write these in shorter notation? 2

• For repeated addition we use multiplication 3+3+3=5× 3 • For repeated multiplication we use exponents 3× 3× 3= 3

Write these numbers using exponential notation: • 10 000 = 10? • 27 = 3? 104 33 • 32 = 2? 25 4

Computer Memory A byte is capable of storing one letter of the alphabet. For example, the word “math” requires four bytes to store in a computer. Bytes of computer memory are often manufactured in amounts equal to powers of 2. 5

For Example 1 kilobyte (1 Kb) = 210 = 1024 (about a thousand bytes) 1 megabyte (1 Mb) = 220 = 1 048 576 (about a thousand kilobytes) 1 gigabyte (1 Gb) = 230 = 1 073 741 824 (about a thousand megabytes) 6

Integer Exponents • Base 4 • Exponent 3 • 43 is called a power • 43 = 4 x 4 • = 64 You need to know what is meant by base, exponent and power. 7

You try these. . . • 72 = 49 • 4 5 = 625 • 3 4 = 64 8

Exponent Rules 1. 5. 2. 6. 3. 7. 4. These rules will be explained in the following slides 9

Exponent Rule #1 #1 Why put a 1 at the beginning? n times 5 times That’s why! 10

Exponent Rule #2 #2 Any power with an exponent of 0 is equal to 1 11

0 a = 1 • 0 4 =1 • M 0 =1 • 0 9 =1 • (pq)0 =1 • 170 =1 • (2 x 2 y)0 =1 12

Exponent Rule #3 #3 n times 4 times A negative exponents means divide 13

Negative Exponents • -1 3 • -4 2 • 7 -1 • (1/2) -1 • 5 -2 • (3/4) -2 14

Operations with Exponents Multiply: (x 3)∙(x 4) = (x ∙ x) ∙ (x ∙ x ∙ x) = (x ∙ x ∙ x ∙ x) = x 7 A 5 ∙ A 4 = A 9 Divide: = m 2 16

Multiplying Exponents m n a a =? 17

Exponent Rule #4 #4 When multiplying powers with the same base, add the exponents. 18

• x 5 x 4 = x 9 • y-2 y 7 = y 5 • b 3 b-6 = b-3 • m 6 m-6 = m 0 • (2 x 5)(4 x-3) • 2 z-1 z 5 w-6 w-2 = 8 x 2 = 2 w-8 z 4 19

Dividing Exponents 20

Exponent Rule #5 #5 When dividing powers with the same base, subtract the exponents. 21

= x 4 = y 5 22

Powering Exponents (am)n=? 23

Exponent Rule #6 #6 When taking a powers of a power (one base only), multiply the exponents. 24

• (w 4)5 = w 20 • (p-3)4 = p-12 • (5 x 4)-2 = 5 -2 x-8 • (-3 y-7)3 = -33 y-21 25

Powering Exponents 26

Exponent Rule #7 #7 Powers of products or quotients When taking a powers of a product or quotient, distribute the exponent into the bracket. 27

Exponent Rules 1. 5. 2. 6. 3. 7. 4. 29