An index, or power, is the small floating number that appears after a number or letter. Indices show how many times a number or letter has been multiplied by itself.
An example of a fractional index is \(g^{\frac{1}{3}}\). The denominatorThe bottom part of a fraction. For ⅝, the denominator is 8, which represents 'eighths'. of the fraction is the rootThe root of an equation is the same as the solution to the equation. of the number or letter, and the numeratorThe top part of a fraction. For ⅝ , the numerator is 5. of the fraction is the power to raise the answer to.
By using the multiplication rules from earlier it is clear to see that:
\(g^{\frac{1}{2}} \times g^{\frac{1}{2}} = g^1\)
Therefore: \(g^{\frac{1}{2}} = \sqrt{g}\)
In general, \(a^{\frac{1}{2}} = \sqrt{a}\), \(a^{\frac{1}{3}} = \sqrt[3]{a}\) and so on.
Example
\(8\frac{1}{3} = 2\)
It is possible to combine fractional indices with raising a power to a power.
\(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\)
Question
Write \(t^{\frac{3}{2}}\) in root form.
Simplify \(8^{\frac{2}{3}}\).
\(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\), so \(t^{\frac{3}{2}} = (\sqrt[2]{t})^3\)