Exponents

Negative Exponents Key idea: turn the exponent into a positive, then simplify. Opposite place, opposite sign. Turn the exponent into a fraction. Next, flip. When you flip, the negative exponent becomes positive. Example: 9⁻²  9⁻²/1  1/9²  1/81 When you have an unknown, it cannot have a value that will cause the denominator to be zero. We can’t divide by zero.

Multiplying Numbers with the Same Base If a number is multiplied by another number with the same base, you add the exponents. Example: 2³ x 2⁴ = (2 x 2 x 2)(2 x 2 x 2 x 2) = 2³⁺⁴ = 2⁷ = 128 2³ x 2⁴ = (8)(16) = 128

Dividing Numbers with the Same Base If a number is divided by another number with the same base, you can subtract the exponents. Example: 2⁵ ÷ 2² = 2⁵¯² = 2³ = 8

Zero as an Exponent Anything with a zero exponent equals 1. Examples: 10⁰ = 1 10² ÷ 10² = 10²¯² = 10⁰ = 1

Raising a Power to a Power When raising an exponent to another exponent, you fast add or multiply the exponents. Example: (6⁷)³ = 6⁷˚³ = 6^21

Simplifying an Exponential Expression There are two techniques that can be used to simplify an exponential expression. The first is to have everything in the numerator. (# on top) The second is to make all the exponents positive.

Examples: Put everything in the numerator: X³X¯² = X³X¯²X¯²X¯⁴ = X¯⁵ Make all the exponents positive: X³X¯² = X³ = X³ = X¯⁵ = 1 X²X⁴ X²X⁴X² X⁸ X⁵