Relate fractional indices to roots

Roots and fractional indices

For positive bases, fractional indices connect roots and powers. Fractional-index operations are optional extension content here; rational and irrational real numbers are part of Year 9.

1. A root reverses a power For a positive base, the index shown below means the positive nth root, where n is an integer at least 2. It does not mean divide the base by n. \(\displaystyle a^{1/n}=\sqrt[n]{a}\)
2. Denominator first, numerator next Here m is a positive integer. Take the root named by denominator n, then raise that result to power m. All bases here are positive \(\displaystyle a^{m/n}=(\sqrt[n]{a})^m\)
3. A negative index takes a reciprocal Keep the root and power calculation, then take its reciprocal. The value is still positive \(\displaystyle a^{-m/n}=\frac{1}{a^{m/n}}\)
4. An expression and an equation differ The square-root symbol denotes one nonnegative value. In contrast, the equation shown below has two real solutions \(\displaystyle \sqrt{49}=7,\qquad x^2=49\Rightarrow x=\pm7\)
5. Classify an exact square root The square root of a nonnegative integer is rational exactly when that integer is a perfect square. A non-square positive integer has an irrational square root. A finite decimal display is an approximation, not a proof of rationality

Worked example

Evaluate the negative fractional power exactly

1. \(\displaystyle \sqrt[4]{81}=3\)The denominator 4 asks for a fourth root
2. \(\displaystyle 81^{3/4}=3^3=27\)The numerator 3 asks for a cube of that root
3. \(\displaystyle 81^{-3/4}=\frac{1}{27}\)The negative index takes the reciprocal

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