Rational Exponents
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Rational exponents allow radicals to be written as fractional exponents. To write a radical with an index of n as a rational exponent, the index becomes the denominator and the exponent of the radicand becomes the numerator. The rules of exponents still apply to rational exponents. Expressions with rational exponents are simplified by having no negative exponents, no fractional exponents in the denominator, not being a complex fraction, and having the least possible index for any remaining radicals. Examples show how to evaluate, simplify, and perform operations on expressions with rational exponents.
Rational Exponents
- 2. Review of Exponential Rules
- 3. Simplifying Expressions with Rational Exponents Radicals can also be expressed as a rational (or fractional) power of an expression. It will sometimes be easier to use this new method of expressing a radical to simplify a radical expression. 1 b = b n n When you see a radical expression, you can convert it to a fractional power.
- 4. Example 1 Write each expression in Radical Form 1 6 1) x 6 x 2 3 2 2) m 3 m
- 5. Example 2 Write each radical using Rational Exponents 1 5 1) b b 5 1 5 7 5 7 2) 3 6x y 6 x y 3 3 3
- 6. Example 3 Evaluate Each Expression −1 1 1) 49 2 7
- 7. Rational Exponents For any nonzero Real number a and any integers m and n, with n > 1 m a = ( a) = a n n m n m
- 8. Notice: The index of the radical becomes the denominator of the rational power, and the exponent of the radicand (expression inside the radical) becomes the numerator. Look at these examples: (1) (2) (3) power root x power =x ro ot
- 10. Example 4 Evaluate 2 (−125) 3 ( − 125 ) 3 2 25
- 11. Example 5 Evaluate −3 36 2 49 343 216
- 12. Simplify (3a )(− 7a ) Example 6 3 1 2 5 17 − 21a 10
- 14. Simplify each expression completely =
- 15. Expression with Rational Exponents An expression with Rational Exponents is simplified when: 1. It has no negative exponents 2. It has no Fractional Exponents in the denominator 3. It is not a Complex Fraction 4. The index of any remaining radical is the least number possible.
- 16. Simplifying radicals is often Simplify: easier using rational exponents. 3 Look at this "rational" example, 3 solved two ways. ==> 3 Solved by Rationalizing the Solved by Using Denominator Rational Exponents
- 17. Example 7 Simplify (8x y ) 3 4 2 −1 3 3 1 1 x 4 y 3 1 2 = 2x 4 y 3 2 xy
- 18. Example 8 Simplify −7 4 n 3 n 7 n
- 19. Example 9 Simplify 2 17 n n 12 5 =n 3 1 = 12 n 4 n
- 20. Example 8 Simplify Multiply Change to Same Base Add Exponents Exponents Subtract Exponents 4 2 5 = 4
- 21. Example 9 Simplify 2 a −3 − a +1 a −2 x ⋅x x 2 a −4 = (x ) x 2a −8 =x (a – 2) – (2a – 8) =x -a + 6
- 22. Example 10 Simplify 1 y +1 2 1 y −1 2 1 y + 2y 2 +1 y −1