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Fractional Exponents and Radicals

Fractional Exponents and Radicals. Expressing powers with rational exponents as radicals and vice versa. Today’s Objectives. Students will be able to demonstrate an understanding of powers with integral and rational exponents Explain, using patterns, why a 1/n = n √a , n > 0

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Fractional Exponents and Radicals

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  1. Fractional Exponents and Radicals Expressing powers with rational exponents as radicals and vice versa

  2. Today’s Objectives • Students will be able to demonstrate an understanding of powers with integral and rational exponents • Explain, using patterns, why a1/n = n√a, n > 0 • Express powers with rational exponents as radicals and vice versa • Identify and correct errors in a simplification of an expression that involves powers

  3. Multiplying Powers • We have learned in the past that: • am*an= am+n • We can extend this law to powers with fractional exponents with numerator 1: • 51/2 * 51/2 = 51/2 +1/2 • =51 • =5 • And: • √5 * √5 = √25 • = 5 • So, 51/2 and √5 are equivalent expressions

  4. Multiplying Powers • Similarly, 51/3 * 51/3* 51/3 = 51/3+1/3+1/3 • = 51 • = 5 • And: • 3√5 * 3√5 * 3√5 • = 3√125 • = 5 • So, 51/3 = 3√5 • These examples indicate that: • Raising a number to the exponent ½ is equivalent to taking the square root of the number • Raising a number to the exponent 1/3 is equivalent to taking the cube root of the number, and so on

  5. Powers with Rational Exponents with Numerator 1 • When n is a natural number and x is a rational number: x1/n = n√x

  6. Example 1: Evaluating Powers of the form a1/n • Evaluate each power without using a calculator: • 271/3 • 271/3 = 3√27 • = 3 • 0.491/2 • 0.491/2 = √0.49 • = 0.7 • Your turn: • (-64)1/3 • (4/9)1/2 • -4, 2/3

  7. Powers with decimal exponents • A fraction can be written as a terminating or repeating decimal, so we can interpret powers with decimal exponents; for example: • 0.2 = 1/5, so 320.2 = 321/5 • What about when the numerator is greater than 1? • To give meaning to a power such as 82/3, we extend the exponent law (am)n = amn so that it applies when m and n are rational numbers.

  8. Numerators greater than 1 • We write the exponent 2/3 as 1/3 * 2, or as 2 * 1/3 • So: • 82/3 = (81/3)2 = (3√8)2 • Take the cube root of 8, then square the result • 3√8 = 2, 22 = 4 • Or: • 82/3 = (82)1/3 = 3√82 • Square 8, then take the cube root of the result • 82/3 = 3√64 = 4

  9. Powers with rational exponents • These examples illustrate that the numerator of a fractional exponent represents a power and the denominator represents a root. The root and power can be evaluated in any order • When m and n are natural numbers, and x is a rational number: • xm/n = (x1/n)m = (n√x)m • And: • xm/n = (xm)1/n = n√xm

  10. Example 2: Rewriting Powers in Radical and Exponent Form • Write 402/3 in radical form in 2 ways • Use am/n = (n√a)m or n√am • 402/3 = (3√40)2 or 3√402 • Your turn: • Write √35 and (3√25)2 in exponent form • 35/2 and 252/3

  11. Review Powers with Rational Exponents with Numerator 1 • When n is a natural number and x is a rational number: x1/n = n√x Powers with rational exponents • When m and n are natural numbers, and x is a rational number: • xm/n = (x1/n)m = (n√x)m • And: • xm/n = (xm)1/n = n√xm

  12. Homework • Pg. 227 - 228 • 3, 5, 7, 9, 11, 15, 17-21

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