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Rational Exponents and Radicals

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Intermediate Algebra Lessons
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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn
  • about rational exponents
  • how to evaluate rational exponents
  • how to evaluate rational exponents with negative coefficients
  • how to use the law of exponents for rational exponents

Rational Exponents

Any radical can also be expressed as a rational exponent. For example, a cube root is equivalent to an exponent of 1/3; a fourth root is an exponent of 1/4. When using this method to simplify roots, we need to remember that raising a power to a power multiplies the exponents. This topic is important when finding derivatives and in integral calculus.
How to understand the correlation between radicals and fractional exponents? Rational Exponents
Part 1 covers the meaning of a fractional exponent when the numerator is 1, and provides examples. Part 2 covers the meaning of a fractional exponent, and provides examples.

Evaluating Rational Exponents

Rational exponents indicate two properties: the numerator is the base's power and the denominator is the power of the root. When evaluating rational exponents it is often helpful to break apart the exponent into its two parts: the power and the root. To decide if it is easier to perform the root first or the exponent first, see if there exists a whole number root of the base; if not, we perform the exponent operation first.

How to evaluate a fractional exponent? Evaluating Numbers with Rational Exponents by using Radical Notation

Evaluating Numbers Raised to Fractional Exponents

Rational Exponents with Negative Coefficients

A negative coefficient of a term with a rational exponent can mean that we either (1) apply the rational exponent and then take the opposite of the result, or (2) the rational exponent applies to a negative term. In case 2 of rational exponents with negative coefficients, the answer will be not real if the denominator of the exponent is even. If the root is odd, the answer will be a negative number.

Rules for Rational Exponents

The rules for multiplying and dividing exponents apply to rational exponents as well - however the operations will be slightly more complicated because of the fractions. Some basic rational exponent rules apply for standard operations. When multiplying exponents, we add them. When dividing exponents, we subtract them. When raising an exponent to an exponent, we multiply them. If the problem has root symbols, we change them into rational exponents first.

How to simplify an expression with rational exponents. Using exponent rules with rational exponents

Simplify Fractional Exponents Using the Laws of Exponents Part 1
Simplify Fractional Exponents Using the Laws of Exponents Part 2

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