Exponents

Review: Evaluate each expression: 1.3 ³ 2.7¹ 3.-6² 4.(⅜)³ 5.(-6)² 6.3· 2³ Answers /

Definitions: Exponential Expression: 5⁹ 5⁹= 5(5)(5)(5)(5)(5)(5)(5)(5) Base: the repeated factor of an exponential factor Exponent: number of times that the base is used as a factor – POWER

Review Evaluate the expression by substituting the number in for the variable X = -3 y = 4 z = -1 2x ²y = _________ Answer: 2(-3) ²(4) = 72 9 Z ²x 9 (-1)²(-3) 9/(-3) = -3

What does x ³? X³ = (x)(x)(x)

Product Rule for Exponents If m and n are positive integers, and b is a real number, then….. Example: 3 ³· 3⁴ = 3⁷ Why? (3·3·3) · (3·3·3·3) = 3⁷

Examples: 1. 4 ³· 4 = 2.(-3) ²· (-3) ⁷ = 3.x ³ · x · x ² = 4. 3x³ · 4x² = 5. (3y ⁴)(-10x⁷)(4xy³) = 1.4⁴ 2.(-3)⁹ 3.x⁶ 4.(3 · 4) ( x³ ·x²) = 12x ⁵ 5.(3· -10 · 4)( y ⁴ · y³)(x⁷ · x) = -120y⁷x⁸

Helpful Hint – p. 299 Multiplication: 5x ² · 4x² = 20x ⁴ 6x³ · 7x² = 42x ⁵ Addition: 5x ² + 4x² = 9x² 6x³ + 7x² = Cannot simplify! Not like terms!

Think of your own rule: Simplify each one

Quotient Rule for Exponents If m and n are positive integers, and b is a real number, then…..

Let’s think about it! (x ²)³ = ( x ²)( x ²)( x ²) = x⁶x⁶

Shortcut: Power Rule for Exponents If m and n are positive integers, and b is a real number, then……. Example: (y⁴) ² = y⁸y⁸

Practice: 1.(9⁶)⁸ = 2.(x⁹)⁷ = Think about it: (2x⁵y⁴) ³ Remember: (2x⁵y⁴)(2x⁵y⁴)(2x⁵y⁴) (2 · 2 · 2)(x⁵ · x⁵· x⁵)(y⁴ · y⁴ · y⁴)

Short Cut: Power of a Product Rule If n is a positive integer and a and b are real numbers, then….. Examples: 1). (3y)⁴2). (-2g ²hj ³)⁴

Power of a Quotient Rule If m is a positive integer and b and c are real numbers, then…..

Example:

Why are these two the same? x⁴ 5555

Zero Exponent Rule If a is a real number, then… As long as a is not zero!

Practice 1.x⁰ 2.7⁰ 3.(-3)⁰ 4.-3⁰ 5.x⁵y⁰ x⁵