Indices and standard form

• Use positive, zero, negative and fractional indices: 7⁻² = 1/49, 81^1/2 = 9, 8^−2/3 = 1/4.
• Use the rules of indices to multiply, divide and raise powers.
• Write numbers in standard form A × 10ⁿ, where 1 ⩽ A < 10 and n is an integer, and convert back.
• Calculate with numbers in standard form, with and without a calculator.

Same base: add, subtract or multiply the powers

These rules only work when the base is the same.

What the index means

Any non-zero number to the power 0 is 1. A negative index means “one over” (the reciprocal). A fractional index means a root: the denominator is the root, the numerator is the power. Take the root first so the numbers stay small.

A × 10ⁿ with 1 ⩽ A < 10

Place the decimal point after the first non-zero digit to get A, then count how many places it moved. Large numbers have a positive power; numbers between 0 and 1 have a negative power.

Multiply and divide the parts separately

Without a calculator: multiply (or divide) the A parts, use the index rules on the powers of 10, then adjust so A is between 1 and 10. To add or subtract, first write both numbers with the same power of 10 (or as ordinary numbers).