Laws Of Indices Calculator

Work out laws of indices instantly with clear inputs, formula shown and shareable results.

About Laws Of Indices Calculator

Beyond the three basic laws, powers distribute over multiplication and division of bases, negative exponents give reciprocals, and a unit-fraction exponent is a root.

Formulas

Distributing over a product

(bc)ᵐ = bᵐ · cᵐ

Negative exponent

b⁻ᵐ = 1 / bᵐ

Fractional exponent

b^(1/m) = ᵐ√b

Worked example

With the default values below, Laws Of Indices Calculator returns (bc)ᵐ = bᵐcᵐ of 1,296.000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Base bminimum 0: 3
Exponent m: 4
Second base cminimum 0: 2

Results

(bc)ᵐ = bᵐcᵐ: 1,296.000000
(b/c)ᵐ = bᵐ/cᵐ: 5.06250000
b⁻ᵐ = 1/bᵐ: 0.0123456790
b^(1/m) — the mth root: 1.31607401

Step by step

1Values used
2Distributing over a product
3Negative exponent
4Fractional exponent
5(bc)ᵐ = bᵐcᵐ= 1,296.000000
6(b/c)ᵐ = bᵐ/cᵐ= 5.06250000
7b⁻ᵐ = 1/bᵐ= 0.0123456790
8b^(1/m) — the mth root= 1.31607401

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For b = 3, m = 4, c = 2?

(6)⁴ = 1296, (1.5)⁴ = 5.0625, 3⁻⁴ = 1/81 ≈ 0.0123457 and ⁴√3 ≈ 1.3160740.

Does (b + c)ᵐ = bᵐ + cᵐ?

No. Powers distribute over multiplication, never over addition — that is one of the most common algebra mistakes.