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**How do we handle fractional exponents?**

Do Now: 28 = 256 Fill in the appropriate information 27 = 2? = 128 26 = 64 2? = 25 = 32 24 = 2? = 16 23 = 8 22 = 4 21 = 2 2-1 = 1/2 2-2 = 2? = 1/4 2-3 = 2? = 1/8

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**How do we handle fractional exponents?**

Do Now: Simplify/Rationalize:

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**Properties of Exponents**

Product of Powers Property am • an = am+n Power of Power Property (am)n = am•n Power of Product Property (ab)m = ambm Negative Power Property a-n = 1/an, a 0 Zero Power Property a0 = 1 Quotients of Powers Property Power of Quotient Property

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**Indices, Exponents, and New Power Rules**

Product of Powers Property am • an = am+n example: 82 • 83 = = 85 example: x3 • x6 = x3 + 6 = x9 Power of Product Property (ab)m = am • bm example: (2 • 8)2 = 22 • 82 example: (xy)5 = x5 • y5 Power of Quotient Property example:

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**beware! -23/2 is not the same as (-2)3/2**

Types of exponents Positive Integer Exponent an = a • a • a • • • • a n factors Zero Exponent a0 = 1 Negative Exponent -n Rational Exponent 1/n Rational Expo. m/n - m/n Negative Rational Exponent beware! /2 is not the same as (-2)3/2

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**Roots, Radicals & Rational Exponents**

radical sign radicand index of a number is one of the two equal factors whose product is that number Square Root has an index of 2 can be written exponentially as Every positive real number has two square roots The principal square root of a positive number k is its positive square root, If k < 0, is an imaginary number

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**Roots, Radicals & Rational Exponents**

radical sign radicand index Cube Root of a number is one of the three equal factors whose product is that number has an index of 3 can be written exponentially as k1/3 principal cube roots

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**Roots, Radicals & Rational Exponents**

radical sign radicand index of a number is one of n equal factors whose product is that number nth Root has an index where n is any counting number can be written exponentially as k1/n principal odd roots principal even roots

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**Indices and Rational Exponents**

square root - k1/2 • k1/2 = k 1/2 + 1/2 = k1 = k 21/2 • 21/2 = 2 1/2 + 1/2 = 21 = 2 cube root - k1/3 • k1/3 • k1/3 = k 1/3 + 1/3 + 1/3 = k1 = k 21/3 • 21/3 • 21/3 = 2 1/3 + 1/3 + 1/3 = 21 = 2 nth root - = k1/n k1/n • k1/n • k1/n = k 1/n + 1/n + 1/n = k1 = k n times n times 81/3 • 81/3 • 81/3 = 8 1/3 + 1/3 + 1/3 = 81 = 8

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Fractional Exponents Radicals Fractional Exponents (ab)m = am • bm

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**Simplifying = x10 - 4 = x5 - 2 = x6 = x3 positive integer exponent**

rational exponent multiplication law simplify & power law division law = x10 - 4 = x5 - 2 = x6 = x3

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**Simplifying – Fractional Exponents**

A rational expression that contains a fractional exponent in the denominator must also be rationalized. When you simplify an expression, be sure your answer meets all of the given conditions. Conditions for a Simplified Expression It has no negative exponents. It has no fractional exponents in the denominator. It is not a complex fraction. The index of any remaining radical is as small as possible.

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Model Problems Rewrite using radicals: Rewrite using rational exponents: Evaluate:

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**Evaluate a0 + a1/3 + a -2 when a = 8**

Evaluating Evaluate a0 + a1/3 + a -2 when a = 8 / replace a with 8 1 + 81/ x0 = 1 x1/3 = x–n = 1/xn 8–2 = 1/82 = 1/64 /64 3 1/64 combine like terms If m = 8, find the value of (8m0)2/3 (8 • 80)2/3 replace m with 8 (8 • 1)2/3 x0 = 1 (8)2/3 = 4

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**Simplifying – Fractional Exponents**

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