1.5 Powers, roots and laws of indices

Key Learning Statements

• Index notation is a way of writing repeated multiplication. For example, you can write 2 × 2 × 2 as 23. 2 is the base and 3 is the index that tells you how many times 2 is multiplied by itself.

• The x√ n of a number is the value that is multiplied by itself x times to reach that number.

• Any number to the power of 0 is equal to 1: a0 = 1.

• Negative indices are used to write reciprocals. a–m is the reciprocal of am because a–m × am = 1.

• To multiply numbers with the same base you add the indices. In general terms am × an = am + n

• To divide numbers with the same base you subtract the indices. In general terms am an = am − n .

• To raise a power to another power you multiply the indices. In general terms (am) n = amn 1 Calculate.

Key Concepts

• Calculating with squares, square roots, cubes, cube roots and other powers and roots of numbers.

• The meaning of zero and negative indices.

• The laws of indices.

Tip

3 Find all the square and cube numbers between 100 and 300.

4 Which of the following are square numbers and which are cube numbers?

625, 128

If you don’t have a calculator, you can use the product of prime factors to find the square root or cube root of a number.

5 Simplify.