Warm-up, 3/28 Compute: 1. 2 x 2 x 2 x 2 = 2. 3 x 3 x 3 = 3. 2 x 2 x 3 x 3 x 3 = 4. 5 x 5 x 2 x 2 = 5. 2 x 2 x 4 =

Exponents

Review: What do exponents tell us? What does 5 3 mean? 5 3 =5  5  5

What is the “base”? What is the “exponent”? Base Exponent

Write 6 3 in standard form. 4 3 = 4  4  4 = 64

Write 6 4 in standard form. 3 4 = 3  3  3  3 = 81

What happens if we multiply 2 3  2 4 ? 2 3  2 4 = (2  2  2)  (2  2  2  2)

So… 2 3  2 4 = 2 7 because we used 2 as a factor seven times! Notice anything?

Let’s try another example: What is 5 2  5 4 ? 5 2  5 4 = (5  5)  (5  5  5  5) or 5 2  5 4 = 5 6

For MULTIPLICATION: x s  x t = x s + t (For any number x and for integers s and t.)

In other words, when multiplying two numbers in exponential form with the same base, add the exponents.

Ex: 9 4  9 8 = ? 9 12

What happens if we divide? = 7  7  7  7   7  7 Cancel out any numbers that are in the numerator AND denominator!

After we cancel, we are left with 7 2. So, 7 5 =

For DIVISION: x s = x t (For any nonzero number x and for integers s and t.) x s - t

In other words, when dividing two numbers in exponential form with the same base, subtract the exponents.

Ex: 6 8 ÷ 6 5 = ? 6 3

What about (5 2 ) 3 ? (5 2 ) 3 (5 2 )  (5 2 )  (5 2 ) (5  5)  (5  5)  (5  5) 5 6

Raising a power to a power: (x s ) t = x s  t (For any number x and for integers s and t.)

In other words, when raising a number with an exponent to a power, multiply the exponents.

Ex: (5 3 ) 7 = ? 5 21

One important thing to remember… x 0 = 1 (Any number to the zero power is equal to 1!)

Why??? For example… 3 2 ÷ 3 2 = = = 3  3 = 9 =  3 9

So, 3 0 = 1 This works for any base number!

Can we have negative exponents? Yes! Negative exponents represent the reciprocal. The reciprocal of x is The reciprocal of 10 is They do NOT make the answer negative.

5 -2 = = 2 -4 =

Review: Compute: 2 5 Answer: 32

Compute: 5 -2 Answer: 1/25

Compute: 0 2 Answer: 0

Compute: 2 0 Answer: 1

Compute: Answer: -16

Compute: (-4) 2 Answer: 16

Compute: Answer: - 1/16

Compute: 3 -2 Answer: 1/9

Compute: (2 2 )(2 3 ) Answer: 2 5 or 32

Compute: (3 3 ) 2 Answer: 3 6 or 729

Multiply and express in exponential form: (4 5 )(8 2 ) Answer: 2 16

What is one-third of 3 99 ? Answer: 3 98