3.1 Powers and roots — Learn

Prerequisites: multiplication, division and square numbers.

Objectives: calculate integer powers; find square, cube and higher roots; connect a root to the power that reverses it.

By the end of this section, you can calculate integer powers, find roots, and check a root by using its matching power.

This section covers N6.

Powers

A power is a short way to write repeated multiplication. In $3^4$, the base is $3$ and the index is $4$. It means multiply four copies of 3.

Example 1. Work out $2^5$.

The index tells you how many factors to multiply. It does not tell you to multiply the base by the index.

Roots

A square root reverses squaring. The square root of 49 is 7 because $7^2=49$. The symbol $\sqrt{\ }$ means the positive square root.

A cube root reverses cubing. The cube root of 27 is 3 because $3^3=27$.

Example 2. Work out $\sqrt{144}$ and $\sqrt[3]{125}$.

To check a root, raise your answer to the matching power.

Watch out. $\sqrt{49}$ means the positive root, 7. The equation $x^2=49$ has two solutions, $x=7$ and $x=-7$, because both square to 49. Check whether the question asks for a root or for solutions to an equation.

Progress check

The file doubles once per round. After 3 rounds it has $2^3=8$ units. Predict how the power will represent the size after 20 rounds.

You should now be able to

• write repeated multiplication as a power and calculate it;
• find square, cube and higher roots of known powers;
• check a root by raising it to the matching power;
• distinguish a positive square root from the two solutions of a square equation.