Laws of indices calculator (powers and roots)
Enter a base and two indices and this calculator applies all three index laws at once: add the powers when multiplying, subtract when dividing, and multiply when raising a power to a power. It also evaluates each result and explains negative and fractional indices in words.
Enter a base and two indices
2^5 × 2^3 = 2^8
Value: 256
2^5 ÷ 2^3 = 2^2
Value: 4
(2^5)^3 = 2^15
Value: 32768
Working
- Multiplying: add the indices, 5 + 3 = 8.
- Dividing: subtract the indices, 5 − 3 = 2.
- Power of a power: multiply the indices, 5 × 3 = 15.
The method behind it
- Multiplying: add the indices. Dividing: subtract them.
- Power of a power: multiply the indices.
- a^-n = 1 / a^n and a^(m/n) = the nth root of a, to the power m.
- a^0 = 1 for any non-zero base.
How to use this tool
1. Enter the base
Type the number that is being raised to a power, for example 2, 5 or 0.5. The base must be the same for the multiply and divide laws to apply.
2. Enter the two indices
Type m and n. Negative indices and fractions such as 0.5 (a square root) or 1/3 written as 0.3333 are allowed.
3. Compare index form and value
Use the simplified index form for algebra marks and the evaluated number when the question asks for a value.
Frequently asked questions
What are the three laws of indices?
a^m x a^n = a^(m+n), a^m / a^n = a^(m-n) and (a^m)^n = a^(mn). They only apply when the base a is the same in every term.
What does a negative index mean?
A negative index means the reciprocal: a^-n = 1 / a^n. So 2^-3 = 1/8. The minus sign never makes the answer negative.
What does a fractional index mean?
The denominator is a root and the numerator is a power: a^(m/n) is the nth root of a, raised to the power m. So 8^(2/3) = (cube root of 8)^2 = 4.
Why is anything to the power zero equal to 1?
Dividing a^m by a^m gives a^(m-m) = a^0, and any non-zero number divided by itself is 1, so a^0 = 1 for every non-zero a.
Practise this properly, not just once
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