Negative and Fractional Exponents, Worked Through

Exponents behave while they are whole numbers you can count on your fingers. The trouble starts at zero and continues downwards, which is a shame, because the rules underneath stay consistent the whole way.

Negative exponents flip, they do not subtract

2³ is 8, 2² is 4, 2¹ is 2. Each step down halves the value. Carry on and 2⁰ is 1, 2^-1 is 0.5, 2^-2 is 0.25. Nothing turned negative. The numbers just got small.

So 3^-2 is 1 divided by 9, or 0.1111, and 10^-6 is one millionth. That is why a micrometre is written as 10^-6 m instead of a decimal point followed by six zeros.

Fractional exponents are roots

The denominator says which root to take and the numerator says which power to raise it to. 8^1/3 is the cube root of 8, which is 2. 8^2/3 is that answer squared, so 4. 16^1/2 is the square root, 4, and 16^3/2 is 4 cubed, 64.

Take the root first whenever you can. 64^2/3 done as root-then-square is 4 squared, or 16, which you can do in your head. Done the other way round it is the cube root of 4,096. Same answer, much worse arithmetic.

The rules that actually get used

Multiply powers of the same base and add the exponents: 2⁵ x 2³ is 2⁸, or 256. Divide and subtract them: 2⁵ divided by 2³ is 2², or 4. Raise a power to another power and multiply: (2³)² is 2⁶, or 64.

Two traps live here. First, -3² is -9 while (-3)² is 9, because the minus sign is not part of the base unless brackets say so. Second, a stacked tower like 2^(3^2) works from the top down: 3² is 9, so the answer is 2⁹, which is 512 and not 64.

Where this turns into money

Compound growth is an exponent wearing a different hat. £5,000 at 6% for 30 years is 5,000 x 1.06³⁰. That multiplier is about 5.7435, so the balance lands near £28,717. Change the exponent to 20 years and the multiplier drops to 3.207, giving £16,036.

Ten years of the same rate on the same money, and the difference is £12,681. Compound interest worked step by step uses the identical exponent with the money vocabulary attached.

How to use the exponent calculator

Enter the base, then the exponent. Negative exponents go in with the minus sign in front. Fractional ones can be typed as a decimal, so 0.5 for a square root and 0.3333 for a cube root, or as a fraction if the field accepts one. For a negative base, put brackets around it, or the sign of the answer will come out wrong.

Very large or very small results come back in scientific notation, where 2.5e-7 means 2.5 x 10^-7. If all you want is a root, the square root calculator guide is more direct than entering 0.5 as a power.

Common questions

Why is anything to the power of zero equal to one? Because 2³ divided by 2³ is obviously 1, and the subtraction rule says the same expression is 2⁰. Both are true at once, so 2⁰ is 1. The argument works for every base except zero itself.

Can an exponent be a decimal like 1.5? Yes, and it means the same as 3/2: root first, then power. 9^1.5 is the square root of 9, cubed, which is 27.

What happens with a negative base and a fractional exponent? Often nothing usable. (-8)^1/3 is -2, but (-8)^1/2 has no real answer at all, and most calculators return an error rather than a complex number.

Is x² the same as 2x? Only when x is 0 or 2. At x = 5 one gives 25 and the other gives 10. It is the most common slip in early algebra and it never entirely goes away.