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Number · Indices

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Zero and negative indices

Why anything to the power zero is 1 (not 0), why a negative index means a reciprocal (not a negative number), and why a fraction raised to a negative power flips.

Number · Indices

Zero and negative indices

Why anything to the power zero is 1 (not 0), why a negative index means a reciprocal (not a negative number), and why a fraction raised to a negative power flips.

Why it works

Zero and negative indices aren't new rules — they're what the division law forces to be true if the pattern of powers is to keep working.

Why a0=1a^0 = 1 (not 0). Divide equal powers:

2323=23−3=20,but also88=1.\frac{2^3}{2^3} = 2^{3-3} = 2^0, \qquad \text{but also} \qquad \frac{8}{8} = 1.

So 202^0 must be 11. You can also see it by walking down the ladder: 23=82^3 = 8, 22=42^2 = 4, 21=22^1 = 2 — each step divides by 2 — so the next step is 20=12^0 = 1. Anything (non-zero) to the power zero is 11.

Why a negative index means a reciprocal (not a negative number). Keep walking down the same ladder, dividing by 2 each time:

21=2,20=1,2−1=12,2−2=14,2−3=18.2^1 = 2, \quad 2^0 = 1, \quad 2^{-1} = \tfrac{1}{2}, \quad 2^{-2} = \tfrac{1}{4}, \quad 2^{-3} = \tfrac{1}{8}.

The minus sign in the index never makes the answer negative — it moves the power downstairs:

a−n=1an,so4−2=116, not −16 or −8.a^{-n} = \frac{1}{a^n}, \qquad \text{so} \qquad 4^{-2} = \frac{1}{16}, \text{ not } -16 \text{ or } -8.

Watch what the index is actually attached to. In 4x−24x^{-2} the index sits on the xx only, so it means 4x2\frac{4}{x^2} — the 4 stays upstairs. Only a bracket puts the whole thing downstairs: (4x)−2=116x2(4x)^{-2} = \frac{1}{16x^2}.

A fraction to a negative power flips. The reciprocal of 25\frac{2}{5} is 52\frac{5}{2}, so

(25)−3=(52)3=1258.\left(\tfrac{2}{5}\right)^{-3} = \left(\tfrac{5}{2}\right)^{3} = \tfrac{125}{8}.

Flip first, then apply the positive power — much cleaner than wrestling with 1÷81251 \div \frac{8}{125}.

Negative indices obey all the usual laws. 4x−3×2x5=8x24x^{-3} \times 2x^5 = 8x^2 (add the indices: −3+5=2-3 + 5 = 2), and 6x22x−3=3x2−(−3)=3x5\frac{6x^2}{2x^{-3}} = 3x^{2-(-3)} = 3x^5 — subtracting a negative index pushes the power up, because dividing by "one over x3x^3" is multiplying by x3x^3.