indices calculator

What are indices?

Indices (also called exponents or powers) tell you how many times a number is multiplied by itself. In the expression 2⁵, the base is 2 and the index is 5, which means 2 × 2 × 2 × 2 × 2 = 32.

Indices are foundational in algebra, science, finance, and computing. Whether you are calculating compound growth, simplifying equations, or converting between standard and scientific notation, strong index skills save time and reduce mistakes.

Core laws of indices you should know

1. Product rule (same base)

When multiplying powers with the same base, add the exponents:

a^m × aⁿ = a^m+n

• Example: 3² × 3⁴ = 3⁶ = 729

2. Quotient rule (same base)

When dividing powers with the same base, subtract exponents:

a^m ÷ aⁿ = a^m−n

• Example: 5⁷ ÷ 5² = 5⁵ = 3125

3. Power of a power

When a power is raised to another power, multiply exponents:

(a^m)ⁿ = a^mn

• Example: (2³)⁴ = 2¹² = 4096

4. Zero index

Any nonzero number raised to the power of zero equals one:

a⁰ = 1, for a ≠ 0

5. Negative index

A negative exponent means reciprocal:

a⁻ⁿ = 1 / aⁿ

• Example: 10⁻³ = 1/1000 = 0.001

6. Fractional index

Fractional exponents represent roots:

a^1/n = ⁿ√a and a^m/n = (ⁿ√a)^m

• Example: 81^1/4 = 3

How to use this indices calculator effectively

• Power Calculator: best for direct evaluation like 7³ or 2.5⁻².
• Law Simplifier: great for algebra practice and checking simplification steps.
• Root Calculator: useful for converting between radical and index notation.

If a result is undefined in real numbers (for example, negative base with non-integer exponent), the tool tells you directly so you can spot domain issues early.

Common mistakes students make with indices

• Adding bases instead of exponents: 2³ × 2⁴ is not 4⁷; it is 2⁷.
• Confusing negative indices with negative values: 3⁻² is positive 1/9, not −9.
• Forgetting index laws require same base for product/quotient rules.
• Assuming all roots of negatives are real: even roots of negative numbers are not real.

Why indices matter in real life

Finance and investing

Compound growth uses powers constantly: A = P(1+r)^t. Even small rate changes can create large differences over time due to exponentiation.

Science and engineering

Scientific notation, radioactive decay, and scaling laws all depend on powers and roots. Understanding indices improves your ability to estimate and compare very large or very small values.

Data and computing

Binary systems, algorithmic growth, and storage units (2¹⁰, 2²⁰, etc.) are index-heavy topics in computer science.

Quick practice set

• 4³ × 4² = ?
• 9⁵ ÷ 9³ = ?
• (5²)³ = ?
• 16^−1/2 = ?

Tip: Work these manually first, then verify using the calculator above.