5.9: Rational Exponents

Exponents are not always integers. This section will look into the cases where an exponent is a rational number. When an exponent is a rational number, the expression may be written as an expression with a radical. The rule is to write your answer in the same form as the original problem (if you start with exponents, end with exponents, or if you start with radicals, end with radicals).

Definition: Rational Exponents of the Form \(\dfrac{1}{n}\)

For any real number \(a\) and any integer number \(n\), an expression with the exponent of \(\dfrac{1}{n}\) may be expressed as the following

\[a^{\frac{1}{n}} = \sqrt[n]{a} \nonumber \]

Note: \(n\) is the index in the radical. \(\sqrt[n]{a}\) is read "the nth root of a"

Note: When the radical does not have a visible index, by default the index is \(2\) (square root). Indices greater than \(2\) will be marked on the radical.

Example 5.9.1

1. \((4)^{\frac{1}{2}} = \sqrt{4} = 2\) \(\text{Index is \(2\) by default}\)
2. \( (x)^{\frac{1}{7}} = \sqrt[7]{x}\) \(\text{Index is \(7\)}\)
3. \((−3y)^{\frac{1}{3}} = \sqrt[3]{(-3y)}\) \(\text{Index is \(3\)}\)

Now, Let’s observe what happens when the exponent is a rational number with numerator \(\neq 1\).

Definition: Rational Exponents of the Form \(\dfrac{m}{n}\)

For any real number \(a\) and any integer number \(n\) and \(m\), an expression with the exponent of \(\dfrac{m}{n}\) may be expressed as the following

\[a^{\frac{m}{n}} = \sqrt[n]{a^m} \text{ or } (\sqrt[n]{a})^m \nonumber \]

Note: \(n\) is the index in the radical and \(m\) is the power of the base.

Example 5.9.2

Write the following in radical form

1. \((x)^{\frac{2}{3}} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2\) \(\text{Index is \(3\) and base is raised to the power of \(2\).}\)
2. \((5t)^{\frac{7}{8}} = \sqrt[8]{5t^7} = (\sqrt[8]{5t})^7\) \(\text{Index is \(8\) and base is raised to the \(7\) power.}\)
3. \((x)^{\frac{2}{3}} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2\) \(\text{Index is \(3\) and base raised to the power \(2\).}\)
4. \(\begin{array} &&(z)^{−\frac{5}{9}} &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Given} \\ &= \dfrac{1}{(z)^{\frac{5}{9}}} &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Negative exponent rule applied} \\ &= \dfrac{1}{\sqrt[9]{x^5}} \text{ or } \left( \dfrac{1}{\sqrt[9]{x}} \right)^5 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Rational exponent written as a radical.} \end{array}\)
5. \(\left( \dfrac{3}{4} \right)^{\frac{5}{7}} = \sqrt[7]{\dfrac{3}{4}^5}\) \(\text{Rational exponent written as radical with index \(7\) and base raised to the power of \(5\).}\)

Exercise 5.9.1

Write the following in radical form.

1. \((x)^{\frac{5}{7}}\)
2. \((xy)^{\frac{9}{8}}\)
3. \((x)^{\frac{9}{5}}\)
4. \((z)^{−\frac{11}{13}}\)
5. \(\left( \dfrac{x}{4} \right)^{\frac{6}{9}}\)
6. \(6(y)^{\frac{1}{17}}\)
7. \((6y)^{\frac{1}{17}}\)
8. \(\left( \dfrac{3}{4} \right)^{\frac{x}{y}}\)
9. \(\left( \dfrac{7}{4} \right)^{(−\frac{x}{y})}\)