Chapter 4 Exponents and Polynomials

The Rules of Exponents Chapter 4.1

The Product Rule x a x b = x a + b

1. Multiply. a. a 12 a a a a a a a a a a a a a a 5 a 7 Count how many you are multiplying. Just add the exponents.

1. Multiply. b.w 10 w w w 11 Add the exponents.

2. Simplify, if possible. a.x 3 x 9 x x 12 Same base, add the exponents.

2. Simplify, if possible. b Same base, add the exponents.

2. Simplify, if possible. c.a 3 b 2 a3b2a3b2 Different bases, can’t use the product rule.

3. Multiply. a.( - a 8 )(a 4 ) - a a 12 Multiply the coefficients. Add the exponents. ( - 1)(1)

3. Multiply. b.(3y 2 )( - 2y 3 ) - 6 y y 5 Multiply the coefficients. Add the exponents. (3)( - 2)

3. Multiply. c.( - 4x 3 )( - 5x 2 ) 20 x x 5 Multiply the coefficients. Add the exponents. ( - 4)( - 5)

( (2)( -  )(6) - 3 (x ) (y ) Multiply. x 4 y 5 Multiply and simplify the coefficients. Add the exponents for x and then for y. 2 )( y x 6 y x2x2 -- y3y3 x )

x a x b = x a + b The Product Rule

xbxb xaxa = x a – b if a > b xbxb xaxa = if b > a x b – a 1 xaxa xaxa = x0x0 = The Quotient Rules

xbxb xaxa = x a – b if a > b 1.

a – 7 5. Divide. Count how many are crossed out. Just subtract the exponents.

b. x 11 – 1 x 10 x x Divide. Higher exponent in numerator. Subtract the exponents.

c. y 18 – 8 y 10 y8y8 y Divide. Higher exponent in numerator. Subtract the exponents.

The Quotient Rules xbxb xaxa = x a – b if a > b xbxb xaxa = if b > a x b – a

a. c4c4 c3c3 c 4 – 3 1 c 1 c c c c c c c 6. Divide. The higher exponent is in the denominator.

b – Divide. The higher exponent is in the denominator.

c. z 21 z 15 z 21 – 15 1 z6z Divide. The higher exponent is in the denominator.

a x 2 1 x 9 – 7 7. Divide. Simplify. x7x7 x9x9 The higher exponent is in the denominator.

b x 11 – 4 7. Divide. Simplify. x 11 x4x4 The higher exponent is in the numerator. - 5x 7

c x 9 – 8 7. Divide. Simplify. x8x8 x9x9 The higher exponent is in the denominator. 2x 1

a. y 10 x7x7 x 7 y x7x7 y 10 – 9 8. Divide. y9y9 The higher exponent is in the denominator. Can’t simplify.

b y 2 -x 2 x 5 – 3 y 8 – 6 8. Divide. x5x5 y6y6 x3x3 y8y8 The higher exponent is in the numerator. Simplify. The higher exponent is in the denominator.

The Quotient Rules xbxb xaxa = x a – b if a > b xbxb xaxa = if b > a x b – a 1 xaxa xaxa = x0x0 =

a = Divide. Same exponent.

b Divide. a4a4 a4a4 Simplify. Same exponent.

a c -5b b 8 – 7 c 5 – Divide. a3a3 b8b8 c4c4 a3a3 b7b7 c5c5 The higher exponent is in the numerator. Simplify. The higher exponent is in the denominator. Same exponent.

x0x0 b x 4 y 8 – Divide. y6y6 x4x4 y8y8 The higher exponent is in the denominator. Simplify. 0 exponent. 2x 4 y 2 1

16a 5 b 7 ( -18 a 3 16 b a 2 b Simplify. )( ab5b5 -6 3a2a2 b4b4 ) Simplify. Multiply. Subtract the exponents. Add the exponents in the numerator. a 5 b 7

The Quotient Rules xbxb xaxa = x a – b if a > b xbxb xaxa = if b > a x b – a 1 xaxa xaxa = x0x0 =

The Power Rules (x a ) b = x a b (x a y b ) c = x a c ( ) ybyb xaxa c x a c y b c =

a.(a 4 ) 3 a 4 3 a 12 (a 4 )(a 4 )(a 4 ) 12. Simplify. Can write it three times. Add 4 three times or multiply the exponents.

b.(10 5 ) Simplify. Multiply the exponents.

c.( - 1) Simplify. Multiply - 1 an odd number (15) of times.

a.(3xy) 3 (3) 3 27 x 1 3 y Simplify. Multiply the exponents. Keep 3 in the parentheses. Evaluate each. x3x3 y3y3

y 1 37 z 1 37 Multiply the exponents. Evaluate each. y 37 z 37 b.(yz) 37

( - 3) 2 9 x Simplify. Multiply the exponents. Keep - 3 in the parentheses. Evaluate each. x6x6 c.( - 3x 3 ) 2

a. 5 x ( ) 3 (5) 3 x x Simplify. Keep 5 in the parentheses. Multiply the exponents. Evaluate.

b. ( ( 4 a ) 2 a b ) 6 Multiply exponents. Evaluate. Use quotient rule and subtract exponents. 16 a 2 a 6 b 6 16 a 4 b 6

( ) Simplify. x3x3 y0y0 z x z2z2 Simplify and use quotient rules. Work inside parentheses. Use power rule and evaluate x 2 z ( ) x 10 z 5

The Power Rules (x a ) b = x a b (x a y b ) c = x a c ( ) ybyb xaxa c x a c y b c =

The Rules of Exponents Chapter 4.1