Laws of indices - EdexcelFractional indices - Higher
Indices are used to show numbers that have been multiplied by themselves. They can be used instead of the roots such as the square root. The rules make complex calculations that involve powers easier.
An example of a fractional index is \(g^{\frac{1}{3}}\). The denominatorThe bottom part of a fraction. For 鈪, 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 鈪 , the numerator is 5. of the fraction is the power to raise the answer to.
By using multiplication rules 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
\(8^{\frac{1}{3}} = \sqrt[3]{8} = 2\)
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 \(t^{\frac{3}{2}} = (\sqrt[2]{t})^3\)
It is possible to have negative fractional indices too.
Example
simplifyA fraction is simplified when there are no more common factors shared by the numerator and denominator. For example, the fraction 8/10 simplifies to 4/5 by dividing the numerator and denominator by the common factor of 2.\(27^{\frac{-2}{3}}\).