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Aim: How do we handle fractional exponents?. 2 8 =. 256. Do Now:. Fill in the appropriate information. 2 ? =. 2 7 =. 128. 64. 2 6 =. 32. 2 ? =. 2 5 =. 16. 2 ? =. 2 4 =. 2 3 =. 8. 2 2 =. 4. 2. 2 1 =. 1/2. 2 -1 =. 2 -2 =. 1/4. 2 ? =. 2 -3 =. 2 ? =.
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Aim: How do we handle fractional exponents? 28 = 256 Do Now: Fill in the appropriate information 2? = 27 = 128 64 26 = 32 2? = 25 = 16 2? = 24 = 23 = 8 22 = 4 2 21 = 1/2 2-1 = 2-2 = 1/4 2? = 2-3 = 2? = 1/8
Aim: How do we handle fractional exponents? Simplify/Rationalize: Do Now:
am • an = am+n Product of Powers Property (am)n = am•n Power of Power Property (ab)m = ambm Power of Product Property Negative Power Property a-n = 1/an, a 0 Zero Power Property a0= 1 Quotients of Powers Property Power of Quotient Property Properties of Exponents
am • an = am+n Product of Powers Property example: example: example: example: Power of Product Property Power of Quotient Property example: Indices, Exponents, and New Power Rules 82 • 83 = 82 + 3 = 85 x3 • x6 = x3 + 6 = x9 (ab)m = am • bm (2 • 8)2 = 22 • 82 (xy)5 = x5 • y5
Positive Integer Exponent an = a • a • a • • • • a n factors 0 a0= 1 Zero Exponent Negative Exponent -n Rational Exponent 1/n Negative Rational Exponent Rational Expo. m/n - m/n Types of exponents beware! -23/2 is not the same as (-2)3/2
index radical sign radicand 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 The principal square root of a positive number k is its positive square root, . Every positive real number has two square roots If k < 0, is an imaginary number Roots, Radicals & Rational Exponents
index radical sign radicand of a number is one of the three equal factors whose product is that number Cube Root has an index of 3 can be written exponentially as k1/3 principal cube roots Roots, Radicals & Rational Exponents
index radical sign radicand 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 Roots, Radicals & Rational Exponents
square root - nth root - cube root - n times n times Indices and Rational Exponents k1/2 • k1/2 = k1/2 + 1/2 = k1 = k 21/2 • 21/2 = 2 1/2 + 1/2 = 21 = 2 k1/3 • k1/3 • k1/3 = k1/3 + 1/3 + 1/3 = k1 = k 21/3 • 21/3 • 21/3 = 2 1/3 + 1/3 + 1/3 = 21 = 2 = k1/n k1/n • k1/n • k1/n . . . = k1/n + 1/n + 1/n. . . = k1 = k 81/3 • 81/3 • 81/3 = 8 1/3 + 1/3 + 1/3 = 81 = 8
Radicals Fractional Exponents Fractional Exponents (ab)m = am • bm
positive integer exponent rational exponent multiplication law simplify & power law division law Simplifying = x10 - 4 = x5 - 2 = x6 = x3
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.
Model Problems Rewrite using radicals: Rewrite using rational exponents: Evaluate:
Evaluate a0 + a1/3 + a -2 when a = 8 80 + 81/3 + 8-2 replace a with 8 1+ 81/3 + 8-2 x0 = 1 x1/3 = 1+ 2 + 8-2 1+ 2 + 1/64 x–n = 1/xn 8–2 = 1/82 = 1/64 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 Evaluating (8)2/3 = 4
