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Number · Indices

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Fractional indices

Why a power of 1/n is an nth root, how m/n means "root, then power", how negative fractional indices chain reciprocal + root + power, and how to solve equations by matching bases.

Number · Indices

Fractional indices

Why a power of 1/n is an nth root, how m/n means "root, then power", how negative fractional indices chain reciprocal + root + power, and how to solve equations by matching bases.

Why it works

Why a1/2a^{1/2} is the square root — not half of aa. The multiplication law forces it:

91/2×91/2=912+12=91=9.9^{1/2} \times 9^{1/2} = 9^{\frac{1}{2} + \frac{1}{2}} = 9^1 = 9.

So 91/29^{1/2} is the number that multiplies by itself to give 9 — that is exactly what 9=3\sqrt{9} = 3 means. It is not 4.54.5: the index halves the power count, not the value. Likewise a1/3=a3a^{1/3} = \sqrt[3]{a} (three copies multiply to aa), and in general a1/n=ana^{1/n} = \sqrt[n]{a}.

am/na^{m/n}: root first, then power. Split the fraction with the power-of-a-power law: mn=1n×m\frac{m}{n} = \frac{1}{n} \times m, so

272/3=(271/3)2=32=9.27^{2/3} = \left(27^{1/3}\right)^2 = 3^2 = 9.

Rooting first keeps the numbers small (323^2, not 7293\sqrt[3]{729}) — the same answer either way, but one of them is doable in your head on Paper 1. The denominator is the root, the numerator is the power. (And 272/327^{2/3} is certainly not 27×23=1827 \times \frac{2}{3} = 18 — an index is never a multiplication.)

Negative fractional indices chain three moves. Read 16−3/416^{-3/4} from the outside in: the minus flips, the 4 roots, the 3 powers:

16−3/4=1163/4=1(161/4)3=123=18.16^{-3/4} = \frac{1}{16^{3/4}} = \frac{1}{\left(16^{1/4}\right)^3} = \frac{1}{2^3} = \frac{1}{8}.

Do the moves one at a time, in writing — reciprocal, root, power — and none of them gets dropped.

Fraction bases: flip first. (259)−1/2\left(\frac{25}{9}\right)^{-1/2} — flip for the minus, then root top and bottom: (925)1/2=35\left(\frac{9}{25}\right)^{1/2} = \frac{3}{5}.

Algebraic terms take the fraction everywhere. (16x8)3/4(16x^8)^{3/4}: the index applies to the 16 and the x8x^8:

163/4×x8×34=8x6.16^{3/4} \times x^{8 \times \frac{3}{4}} = 8x^6.

Solving index equations: match the bases. If two powers of the same base are equal, their indices are equal. To solve 2x+1=8x−12^{x+1} = 8^{x-1}, write 8=238 = 2^3:

2x+1=23(x−1)  ⇒  x+1=3x−3  ⇒  x=2.2^{x+1} = 2^{3(x-1)} \;\Rightarrow\; x + 1 = 3x - 3 \;\Rightarrow\; x = 2.

One base, then one linear equation — the standard grade 8–9 finisher.