## Table of Contents

- What is a Rational Exponent?
- Solving Rational Exponents
- Rational Exponent Equation Examples
- Lesson Summary

Learn about rational exponents. Study the definition of rational exponents, learn how to solve them, and work through examples of rational exponent equations.
Updated: 10/06/2021

- What is a Rational Exponent?
- Solving Rational Exponents
- Rational Exponent Equation Examples
- Lesson Summary

Before exploring rational exponents, it is necessary to understand rational numbers. A **rational number** is any number that can be expressed as a fraction. Even whole numbers, integers, are part of the rational number group because we can express any integer as a fraction by setting it over the number one.

Here are some examples of rational numbers:

Number | Fractional Form |
---|---|

1.5 | 3/2 |

4 | 4/1 |

0.01 | 1/100 |

0.25 | 1/4 |

Any number that can be expressed as a fraction is considered a rational number; however, in practice, rational numbers refer to fractions.

What is a **rational exponent**? A rational exponent occurs when a base number is being raised to a fractional (or rational) power. To better understand the rational definition of exponents and prepare for this lesson, review these keywords associated with these two numbers:

{eq}4^{\frac{1}{3}} and \sqrt[3]{4} {/eq}

Term | Definition | Example from Above |
---|---|---|

Base | Main number being manipluated | 4 |

Exponent | Superscripted power of the base | 1/3 |

Rational Exponent | Exponent in fraction form | 1/3 |

Radical form | A number using a root symbol | √ |

Root number (index) | Superscripted number outside the root symbol | 3 |

Radicand | Base number in a radical number (under the root symbol) | 4 |

Understanding the parts of problems using rational exponents is the first step to solving them.

Solving rational exponents is a matter of rewriting the rational exponent in radical form using these steps:

- Make the numerator of the original rational exponent the new exponent of the base.
- Write the base on the (new) exponent as the radicand (number under the root symbol).
- Make the denominator of the original rational exponent the root number, called the
**index**of the radical. - Simplify the radicand expression by expanding the exponent (if possible).
- Simplify the radical (if possible).

For example, if the rational exponent was {eq}\frac{3}{2} {/eq}, then the radical form of this number would require the square root of a cubed number (2 outside the radical and a 3 as the exponent inside the radical).

The best way to understand how to do something is to work through it. Try these rational exponent equations.

Simplify: {eq}5^{\frac{1}{2}} {/eq}

- Step 1
- Identify the 1 as the new exponent.

- Step 2
- Identify the base as the radicand, 5.

- Step 3
- Identify the index number as 2.

So, the rational number {eq}5^{\frac{1}{2}} {/eq} can be rewritten as {eq}\sqrt{{5}^1} {/eq}

- Step 4
- An exponent of 1 is already as simple as possible and does not need to be explicitly written.

- Step 5
- The square root of 5 is not an integer nor a rational number, thus it can be left as a radical or changed to a decimal rounded to any place value desired.

Final Answer: {eq}5^{\frac{1}{2}} = \sqrt{5} {/eq} or {eq}2.24 {/eq} (rounded to the hundredth place value).

Simplify: {eq}(2m)^{\frac{4}{3}} {/eq}

Following steps 1 through 3, rewrite the number with the numerator, 4, as the exponent of the radicand/base, 2, using an index of 3 on the radical.

{eq}\sqrt[3]{2m^4} {/eq}

- Step 4
- Simplify the exponent by calculating {eq}2\times2\times2\times2=16 {/eq}.

- Step 5
- Simplify the radical: 16 is not a perfect cube, however {eq}m^4 {/eq} can be simplified in this instance.

Final Answer: {eq}m\sqrt[3]{16m} {/eq}

Work backward to rewrite this radical expression as a rational exponent expression: {eq}(\sqrt[5]{x})^3 {/eq}

- Step 1
- Start inside the parenthesis and identify that the exponent for the radicand is 1.

- Step 2
- Identify that the index is 5.

- Step 3
- Rewrite the radical expression as the rational exponent expression of {eq}x^{\frac{1}{5}} {/eq}.

- Step 4
- Remember that when a power is raised to a power the two powers are multiplied: {eq}\frac{1}{5} \times 3=\frac{3}{5} {/eq}

- Step 5
- Final answer: {eq}x^{\frac{3}{5}} {/eq}

Rewrite this radical expression as a rational exponent: {eq}\frac{1}{\sqrt[4]{x^3}} {/eq}

- Step 1
- Identify that the radical number is in the denominator of this fraction, thus only the denominator will be altered.

- Step 2
- Identify the exponent of the radicand as the numerator needed for the rational exponent in the final answer, 3.

- Step 3
- Identify the index of the radical as the denominator needed for the rational exponent in the final answer, 4.

- Step 4
- Rewrite the radical number as a rational exponent expression keeping the expression in the denominator of the original fraction.

{eq}\frac{1}{x^{\frac{3}{4}}} {/eq}

- Step 5
- Using knowledge of positive and negative exponents, it is possible to rewrite this as {eq}x^{\frac{-3}{4}} {/eq}.

Final Answer: {eq}\frac{1}{x^{\frac{3}{4}}} {/eq} or {eq}x^{\frac{-3}{4}} {/eq}

Solving rational exponent problems using a scientific calculator results in a decimal answer.

Here are the steps to use a scientific calculator to solve {eq}5^{\frac{2}{5}} {/eq}

- Step 1
- Press the 5 key.

- Step 2
- Press the variable exponent key {eq}x^y {/eq}.

A small box will appear superscripted to the right of the 5 entered in step 1.

- Step 3
- Press the open parenthesis key, (.

- Step 4
- Enter the fraction as a division problem {eq}2\div5 {/eq}.

- Step 5
- Press the closed parenthesis key, ).

- Step 6
- Press the equal key, =.

Final Answer: 1.90 (rounded to two decimal places)

There are a few mistakes that are common when working with rational exponents.

The most common mistakes occur with negative rational exponents. Remember that a negative exponent is made positive by rewriting the number as its inverse. The exponent stays the same but changes from negative to positive.

For example:

{eq}x^{\frac{-3}{4}} {/eq} becomes {eq}\frac{1}{x^{\frac{3}{4}}} {/eq}

Notice that the rational exponent 3/4 goes from negative to positive when the expression is written in inverse.

The reverse is true as well.

{eq}\frac{1}{x^{\frac{-3}{4}}} {/eq} becomes {eq}x^{\frac{3}{4}} {/eq}

Another common error is forgetting that the basic radical sign is a square root, thus has a standard index of 2, and that all base numbers have a standard exponent of 1 if no exponent is showing.

Thus,{eq}\sqrt{x} {/eq} converts to the rational exponent of {eq}x^{\frac{1}{2}} {/eq}.

**Rational exponents** are powers in the form of **rational numbers**, numbers that can be written as fractions. In math calculations, rational numbers tend to refer directly to fractions even though some rational numbers are not fractions (like 5 and 1.5). A radical number is one that includes a root symbol. Remember that the **index** of a radical is the root number (outside the radical sign) and the radicand is the base number under the radical sign.

{eq}x^{\frac{2}{3}}=\sqrt[3]{x^2} {/eq}

When converting a rational exponent expression to radical form, the base becomes the radicand, the numerator of the fractional exponent becomes the exponent for the radicand, and the denominator of the fractional exponent becomes the index of the radical.

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Frequently Asked Questions

Rational exponents are exponents written in the form of a fraction. For example:

x^1/3

3^4/3

xy^2/3

These fractional exponents are manipulated into radical expressions when solving.

Although any number that can be written as a fraction (such as 4 = 4/1) is considered a rational number, in practice rational numbers are presented as fractions. When an exponent is written in fraction form, it is referred to as a rational exponent. The fractional exponent is the rational exponent fraction.

Rational exponents are called rational because they are rational numbers. A rational number is any number that can be written as a fraction.

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