Fractional indices - Higher

An example of a fractional index is g^{\frac{1}{3}}. The denominator of the fraction is the root of the number or letter, and the numerator of the fraction is the power to raise the answer to.

a^{\frac{1}{2}} = \sqrt{a}, a^{\frac{1}{3}} = \sqrt[3]{a} and so on.

By using index laws for multiplication 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}

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

Question
  1. Simplify t^{\frac{3}{2}}.
  2. Simplify 8^{\frac{2}{3}}.
  1. a^{\frac{m}{n}} = (\sqrt[n]{a})^m, so t^{\frac{3}{2}} = (\sqrt[2]{t})^3
  2. a^{\frac{m}{n}} = (\sqrt[n]{a})^m, so 8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4
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