Multiplying powers of the same base

Adding the indices, derived by counting the repeated factors on each side.

Dividing powers of the same base

Subtracting the indices, derived by cancelling the common repeated factors.

The zero index

Why any non-zero number to the power zero is 1, read off the division law.

Negative indices

A negative index as a reciprocal, making a number small rather than negative.

Raising a power to a power

Multiplying the indices, and telling this apart from multiplying powers.

Powers of a product and a quotient

Applying an index to every factor in a bracket and both parts of a fraction.

A negative index on a fraction

Inverting the fraction first, then applying the positive index.

Index equations and changing the base

Writing both sides as powers of one base, then equating the indices.

Unit fractional indices

The index one over n as the nth root, derived from raising a power to a power.

Higher

Fractional indices

The index m over n as a root and a power, and which one to take first.

Higher

Negative fractional indices

Combining reciprocal, root and power in the order that keeps numbers small.

Higher

Practice

Multiplying powers of the same base

Adding the indices, derived by counting the repeated factors on each side.

Practice

Dividing powers of the same base

Subtracting the indices, derived by cancelling the common repeated factors.

Practice

The zero index

Why any non-zero number to the power zero is 1, read off the division law.

Practice

Negative indices

A negative index as a reciprocal, making a number small rather than negative.

Practice

Raising a power to a power

Multiplying the indices, and telling this apart from multiplying powers.

Practice

Powers of a product and a quotient

Applying an index to every factor in a bracket and both parts of a fraction.

Practice

A negative index on a fraction

Inverting the fraction first, then applying the positive index.

Practice

Unit fractional indices

The index one over n as the nth root, derived from raising a power to a power.

Practice

Fractional indices

The index m over n as a root and a power, and which one to take first.

Practice

Index laws review

Negative and fractional indices.

Mixed review Practice

Simplifying powers review

The multiply, divide and power-of-a-power laws with number bases.

Mixed review