Enter a number and a fractional exponent to calculate the result.
A fractional exponent refers to an exponent that is in the form of a fraction, such as \(x^{\frac{a}{b}}\). In this expression, \(x\) is the base and \(\frac{a}{b}\) is the fractional exponent. A fractional exponent can be rewritten as a root operation. For example: \( x^{\frac{a}{b}} = \sqrt[b]{x^a} \) This means you first raise the base \(x\) to the power of \(a\), and then take the \(b\)-th root of the result.
Steps to Calculate a Fractional Exponent:
Formula: \( x^{\frac{a}{b}} = \sqrt[b]{x^a} \)
Solution:
1. First, calculate \(8^2\):
\( 8^2 = 64\)
2. Then, calculate the cube root of 64 (since 3 is the denominator):
\( \sqrt[3]{64} = 4\)
The final result is:
\( 8^{\frac{2}{3}} = 4\)
When the fractional exponent is negative, you need to take the reciprocal before performing the calculation: \( x^{-\frac{a}{b}} = \frac{1}{x^{\frac{a}{b}}}\)
Solution:
1. First, calculate \(27^{\frac{1}{3}}\):
\( \sqrt[3]{27} = 3 \)
2. Then, take the reciprocal:
\( 27^{-\frac{1}{3}} = \frac{1}{27^{\frac{1}{3}}}\)
3. Finally, calculate the result:
\( \frac{1}{27^{\frac{1}{3}}} = \frac{1}{3} \)
Thus, the value of \(27^{-\frac{1}{3}}\) is \(\frac{1}{3} \).