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Chapter 10 Radicals and Rational Exponents

10.1 Finding Roots 10.2 Rational Exponents 10.3 Simplifying Expressions Containing Square Roots 10.4Simplifying Expressions Containing Higher Roots 10.5Adding, Subtracting, and Multiplying Radicals 10.6Dividing Radicals Putting it All Together 10.7Solving Radical Equations 10.8Complex Numbers 10 Radicals and Rational Exponents

For example, can be written in the following way using the definition above. Rational Exponents 10.2 Evaluate Expressions of the form In this section we will see the relationship between radicals and rational exponents. Converting between these two forms makes it easier to simplify expressions. Denominator is the index of the radical.

Example 1 Write in radical form and evaluate. Solution The odd root of a negative number is a negative number.

Evaluate Expressions of the Form Example 2 Write in radical form and evaluate. Solution a)The denominator of the fractional exponent is the index of the radical, and the numerator is the power to which we raise the radical expression. Use the definition to rewrite the exponent. Rewrite as a radical. Problems b through e on next slide..

Example 2-Continued Write in radical form and evaluate. Solution b) To evaluate, first evaluate, then take the negative of that result. Use the definition to rewrite the exponent. Rewrite as a radical. c) The even root of a negative number is not a real number. d) e)

Evaluate Expressions of the Form

Example 3 Rewrite with a positive exponent and evaluate. Solution The reciprocal of 9 is. The denominator of the fractional exponent is the index of the radical. b) The reciprocal of 81 is. The denominator of the fractional exponent is the index of the radical. c) The reciprocal of is. The denominator of the fractional exponent is the index of the radical.

Be Careful The negative exponent does not make the expression negative! Combine the Rules of Exponents Example 4 Simplify completely. The answer should contain only positive exponents. Solution Multiply exponents. Add exponents. Subtract exponents. Simplify

Example 5 Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents. Solution Multiply exponents. Multiply and reduce. Add exponents. Subtract exponents.

Simplify the expression inside the parentheses by subtracting the exponents. Apply the power rule and simplify. Example 5-Continued Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents. Solution

Convert a Radical Expression to Exponential Form and Simplify Some radicals can be simplified by first putting them into rational exponent form and then converting them back to radicals. Example 6 Rewrite each radical in exponential form, then simplify. Write the answer in simplest (or radical) form. Assume the variable represents a nonnegative real number. Solution a)Since the index of the radical is the denominator of the exponent and the power is the numerator, we can write

Example 7 Solution