How to Solve Rational Exponents?
An exponential expression (also called fractional exponents) of the form \(a^m\) has a rational exponent if \(m\) is a rational number (as opposed to integers). Here, you learn more about solving rational exponents problems.
Rational exponents are exponents of numbers that are expressed as rational numbers, that is, in \(a^{\frac{p}{q}}\), \(a\) is the base, and \(\frac{p}{q}\) is the rational exponent where \(q ≠ 0\).
In rational exponents, the base must be a positive integer. Some examples of rational exponents are: \(2^{\frac{1}{3}}\), \(5^{\frac{5}{9}}\),\(10^{\frac{10}{3}}\).
Related Topics
A step-by-step guide to rational exponents
Rational exponents are defined as exponents that can be expressed in the form of \(\frac{p}{q}\), where \(q≠0\).
The general symbol for rational exponents is \(x^{\frac{m}{n}}\), where \(x\) is the base (positive number) and \(\frac{m}{n}\) is a rational power. Rational exponents can also be written as \(x^{\frac{m}{n}}\) \(=\sqrt[n]{m}\)
- \(\color{blue}{x^{\frac{1}{n}}=\sqrt[n]{x}}\)
- \(\color{blue}{x^{\frac{m}{n}}=\:\left(\sqrt[n]{x}\right)^m\:or\:\sqrt[n]{\left(x^m\right)}}\)
Rational exponents formulas
Let’s review some of the formulas for rational exponents used to solve various algebraic problems. The formula for integer exponents is also true for rational exponents.
- \(\color{blue}{a^{\frac{m}{n}}\times a^{\frac{p}{q}}=a^{\left(\frac{m}{n}+\frac{p}{q}\right)}}\)
- \(\color{blue}{a^{\frac{m}{n}}\div a^{\frac{p}{q}}=a^{\left(\frac{m}{n}-\frac{p}{q}\right)}}\)
- \(\color{blue}{a^{\frac{m}{n}}\times b^{\frac{m}{n}}=\left(ab\right)^{\frac{m}{n}}}\)
- \(\color{blue}{a^{\frac{m}{n}}\div b^{\frac{m}{n}}=\left(a\div b\right)^{\frac{m}{n}}}\)
- \(\color{blue}{a^{-\frac{m}{n}}=\left(\frac{1}{a}\right)^{\frac{m}{n}}}\)
- \(\color{blue}{a^{\frac{0}{n}}=a^0=1}\)
- \(\color{blue}{\left(a^{\frac{m}{n}}\right)^{\frac{p}{q}}=a^{\frac{m}{n}\times \frac{p}{q}}}\)
- \(\color{blue}{x^{\frac{m}{n}}=y⇔x=y^{\frac{n}{m}}}\)
Rational Exponents – Example 1:
solve. \(8^{\frac{1}{2}}\times 8^{\frac{1}{2}}\)
Use this formula to solve rational exponents: \(\color{blue}{a^{\frac{m}{n}}\times a^{\frac{p}{q}}=a^{\left(\frac{m}{n}+\frac{p}{q}\right)}}\)
\(8^{\frac{1}{2}}\times 8^{\frac{1}{2}}\) \(=8^{\left(\frac{1}{2}+\frac{1}{2}\right)}\)
\(=8^{1}=8\)
Rational Exponents – Example 2:
Solve. \(2^{\frac{1}{4}}\times 7^{\frac{1}{4}}\)
Use this formula to solve rational exponents: \(\color{blue}{a^{\frac{m}{n}}\times b^{\frac{m}{n}}=\left(ab\right)^{\frac{m}{n}}}\)
\(2^{\frac{1}{4}}\times 7^{\frac{1}{4}}\) \(=(2\times7)^{\frac{1}{4}}\)
\(=14^{\frac{1}{4}}\)
Exercise for Rational Exponents
Evaluate the following rational exponents.
- \(\color{blue}{25^{\frac{1}{2}}}\)
- \(\color{blue}{81^{\frac{5}{4}}}\)
- \(\color{blue}{(2x^{\frac{2}{3}})(7x^{\frac{5}{4}})}\)
- \(\color{blue}{8^{\frac{1}{2}}\div 8^{\frac{1}{6}}}\)
- \(\color{blue}{(\frac{16}{9})^{-\frac{1}{2}}}\)
- \(\color{blue}{\left(8x\right)^{\frac{1}{3}}\left(14x^{\frac{6}{5}}\right)}\)
- \(\color{blue}{5}\)
- \(\color{blue}{243}\)
- \(\color{blue}{14x^{\frac{23}{12}}}\)
- \(\color{blue}{2}\)
- \(\color{blue}{\frac{3}{4}}\)
- \(\color{blue}{28x^{\frac{23}{15}}}\)
Related to This Article
More math articles
- 6th Grade M-STEP Math Worksheets: FREE & Printable
- HSPT Math FREE Sample Practice Questions
- How to Evaluate Integers Raised to Rational Exponents
- 8th Grade Common Core Math FREE Sample Practice Questions
- 10 Most Common 7th Grade MAP Math Questions
- Full-Length PSAT 10 Math Practice Test-Answers and Explanations
- Best Graphing Calculators for High School Students
- 5th Grade RISE Math Worksheets: FREE & Printable
- Top 10 6th Grade PARCC Math Practice Questions
- The Ultimate ParaPro Math Course (+FREE Worksheets & Tests)
What people say about "How to Solve Rational Exponents? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.