Coaching Centre
Fractional indices
Fractional indices as roots and powers, evaluating expressions like 8^(2/3), and applying index laws with fractional and negative indices.
Meaning
Fractional index
$$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$$Evaluate $32^{\frac{3}{5}}$ and $25^{-\frac{1}{2}}$.
$(\sqrt[5]{32})^3 = 2^3 = 8$
$\dfrac{1}{\sqrt{25}} = \dfrac{1}{5}$
Simplify $(x^{\frac{1}{2}})^4$ and $x^{\frac{1}{2}} \times x^{\frac{3}{2}}$.
$x^2$
$x^{2}$ (add the indices: $\tfrac{1}{2} + \tfrac{3}{2} = 2$)
$a^{rac{1}{2}}$
$\sqrt{a}$
$8^{rac{2}{3}}$
$4$
{x^2}$"]$x^{\frac{2}{3}}$
Evaluate $32^{\frac{3}{5}}$.
$2^3$.
Evaluate $25^{-\frac{1}{2}}$.
$\dfrac{1}{5}$.
Evaluate $8^{\frac{2}{3}}$.
$2^2$.
Simplify $(x^{\frac{1}{2}})^4$.
$\tfrac{1}{2} \times 4$.
Evaluate $9^{\frac{3}{2}}$.
$3^3$.
Evaluate $16^{-\frac{1}{2}}$.
$\dfrac{1}{4}$.
Evaluate $\sqrt[4]{16}$.
$2^4 = 16$.
Simplify $x^{\frac{1}{2}} \times x^{\frac{1}{3}}$. Type like x^(5/6).
$\tfrac{3}{6} + \tfrac{2}{6}$.
New numbers every time. Keep going until you get five in a row.