Rational Exponents, Radicals, and Complex Numbers

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Rational Exponents, Radicals, and Complex Numbers

Rational Exponents, Radicals, and Complex Numbers
CHAPTER 8 Rational Exponents, Radicals, and Complex Numbers 8.1 Radical Expressions and Functions 8.2 Rational Exponents 8.3 Multiplying, Dividing, and Simplifying Radicals 8.4 Adding, Subtracting, and Multiplying Radical Expressions 8.5 Rationalizing Numerators and Denominators of Radical Expressions 8.6 Radical Equations and Problem Solving 8.7 Complex Numbers

8.2 Rational Exponents 1. Evaluate rational exponents.
2. Write radicals as expressions raised to rational exponents. 3. Simplify expressions with rational number exponents using the rules of exponents. 4. Use rational exponents to simplify radical expressions.

Rational exponent: An exponent that is a rational number.
Rational Exponents with a Numerator of 1 a1/n = where n is a natural number other than 1. Note: If a is negative and n is odd, then the root is negative. If a is negative and n is even, then there is no real number root.

Example Rewrite using radicals, then simplify if possible. a. 491/2 b. 6251/4 c. (216)1/3 Solution a. b. c.

continued Rewrite using radicals, then simplify. d. (16)1/4 e. 491/2 f. y1/6 Solution d. e. f.

continued Rewrite using radicals, then simplify. g. (100x8)1/2 h. 9y1/5 i. Solution d. e. f.

General Rule for Rational Exponents
where a  0 and m and n are natural numbers other than 1.

Example Rewrite using radicals, then simplify, if possible.
a. 272/3 b. 2433/5 c. 95/2 Solution a. b. c.

continued Rewrite using radicals, then simplify, if possible. d. e. f.
Solution d. e. f.

Negative Rational Exponents
where a  0, and m and n are natural numbers with n  1.

Example Rewrite using radicals; then simplify if possible.
a. 251/2 b. 272/3 Solution a. b.

continued Rewrite using radicals; then simplify if possible. c. d.
Solution c.

Example Write each of the following in exponential form. a. b.
Solution a. b.

continued Write each of the following in exponential form. c. d.
Solution c. d.

Rules of Exponents Summary
(Assume that no denominators are 0, that a and b are real numbers, and that m and n are integers.) Zero as an exponent: a0 = 1, where a 0. 00 is indeterminate. Negative exponents: Product rule for exponents: Quotient rule for exponents: Raising a power to a power: Raising a product to a power: Raising a quotient to a power:

Example Use the rules of exponents to simplify. Write the answer with positive exponents. Solution Use the product rule for exponents. (Add the exponents.) Add the exponents. Simplify the rational exponent.

Example Use the rules of exponents to simplify. Write the answer with positive exponents. Solution Use the product rule for exponents. (Add the exponents.) Rewrite the exponents with a common denominator of 6. Add the exponents.

Example Use the rules of exponents to simplify. Write the answer with positive exponents. Solution Use the quotient for exponents. (Subtract the exponents.) Rewrite the subtraction as addition. Add the exponents.

Example Use the rules of exponents to simplify. Write the answer with positive exponents. Solution Add the exponents.

Example Use the rules of exponents to simplify. Write the answer with positive exponents. Solution

Example Use the rules of exponents to simplify. Write the answer with positive exponents. Solution

Example Use the rules of exponents to simplify. Write the answer with positive exponents. Solution

Example Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values. a. b. Solution

continued Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values. c. Solution

Example Perform the indicated operations. Write the result using a radical. a. b. Solution a. b.

continued Perform the indicated operations. Write the result using a radical. c. Solution c.

Example Write the expression below as a single radical. Assume that all variables represent nonnegative values. Solution