4.1 4.1 PolynomialsPolynomials

4.1 Natural-Number Exponents

ObjectivesObjectives• Learn the meaning of exponential notation• Simplify calculation by using product rule for

exponents• Simplify calculation by using power rule for

exponents• Simplify calculation by using quotient rule for

exponents

Japanese Sword Japanese Sword MakingMaking

•Japanese sword making •Begins with raw iron (1:15-3:07)•Forged and folded 15 times. (4:34-8:06)

•How many layers does this produce?•215 = 32,768

Meaning of Meaning of Exponential NotationExponential Notation

•Note: a3 = a a aa. 25 = 2 2 2 2 2 = 32

b. (–7)3 = (–7)(–7)(–7) = –343

c. –y5 = –y y y y y

d. -53 = -(5 5 5) = -125

Meaning of Meaning of Exponential NotationExponential Notation

If n is a natural number, then

xn = x x x … x

xn

n factors of x

Base Exponent

ExamplesExamplesa. 31 = 3 b. (–y)1 = –y c. (–4z)2 = (–4z)(–4z) = 16z2

d. (t2)3 = t2 t2 t2 = t6

ExamplesExamples• Show that (–2)4 and –24 have different values.

o (–2)4 = (-2)(-2)(-2)(-2) = 16o –24 = -(2 2 2 2) = -16

ExamplesExamplesSimplify

1. 25

2. (-2)3

3. (-3)4

4. 33

General Rule◦ If the exponent is even, result is positive.◦ If the exponent is odd, result is the same sign as that of the original

base.

Product Rule for Product Rule for ExponentsExponents

We note: x2x3 = x x x x x = x x x x x = x5

Product Rule for Exponents

xm xn = x x x . . . x x x x . . . x = x x x x x x . . . x x x = xm + n

m factors of x n factors of x

m + n factors of x

Product Rule for Product Rule for ExponentsExponents

If m and n are natural numbers {1, 2, 3, …}, then

xmxn = xm + n

Power Rule for Power Rule for ExponentsExponents

If m and n are natural numbers {1, 2, 3, …}, then

(xm)n = xmn

Example

a.(23)7 = 23 7

= 221

b. (z7)7 = z7 7

= z49

ExampleExample• Simplify (2x)3

(2x)3 = (2x)(2x)(2x)

= (2 2 2)(x x x)

= 23x3

= 8x3

ExamplesExamples Simplify each expression.a. x3x4 = x3 + 4

= x7

b. y2y4y = (y2y4)y = (y2 + 4)y = y6y = y6 + 1

= y7

Power Rules for ExponentsPower Rules for ExponentsWe note:

(x3)4 = x3 x3 x3 x3

= x x x x x x x x x x x x = x12

This suggests the Power Rule for Exponents. (xm)n = xm xm xm . . . xm

= x x x x x x x . . . x = xm n

12 factors of x

X3 x3x3 x3

m n factors of x

Your TurnYour TurnSimplify:

a.(23)7 = 23 7

o= 221

b. (z7)7 = z7 7

o= z49

C.(2x)3 = (2x)(2x)(2x)a. = (2 2 2)(x x x)

= 23x3

= 8x3

ExampleExampleSimplify.

Quotient Rule for ExponentsQuotient Rule for Exponents

Note:

This suggests:

= 45-2 = 43

Quotient Rule for ExponentsQuotient Rule for Exponents

Quotient Rule for ExponentsIf m and n are natural numbers, m n and x 0, then

ExamleExamle

Simplify. Assume no division by zero.

ExampleExampleSimplify. Assume no division by zero.