0580 · 1.3Powers and roots

Powers, squares and roots

0:00A square with sides of five units contains twenty five unit squares. Multiplying the side length by itself gives the area. Finding the side length from the area reverses that calculation. Powers and roots describe these two directions, and understanding their connection makes the notation much easier to use. You need multiplication, division, and the rule that multiplying two negative numbers gives a positive result. A positive whole number power means repeated multiplication of the same number.

0:273⁴ means four factors of three multiplied together. It does not mean 3 × 4. The number being repeatedly multiplied is called the base. The small raised number is the index, also called the exponent. Here, three is the base and four is the index. 3 × 3 gives nine, then multiplying by three gives twenty seven, and multiplying again gives eighty one. Squaring means raising a number to the power of two.

0:545² therefore means 5 × 5, giving twenty five. Cubing means raising a number to the power of three. 2³ means 2 × 2 × 2, giving eight. The names square and cube come from geometry. A square uses two equal side lengths when calculating its area. A cube uses three equal dimensions when calculating its volume. Three layers of nine unit cubes give twenty seven altogether, and that total is 3³.

1:23A square root reverses squaring. √25 is five because 5² = 25. The square root symbol means the non-negative square root. Although negative five also squares to twenty five, the symbol by itself gives five, not two answers. A cube root reverses cubing. The cube root of eight is two because 2³ = 8. The small three on the root symbol tells you which power to reverse. Unlike square roots, cube roots of negative numbers are real numbers: the cube root of negative eight is negative two.

Recall facts and worked calculations

1:56For Cambridge, you should recall the squares of the whole numbers from one to fifteen, and their corresponding square roots. Learn each pair in both directions. Knowing that 13² = 169 also tells you its square root. These are connected facts, rather than separate calculations to memorise. You also need the cubes and corresponding cube roots of one, two, three, four, five, and ten. Their cubes are one, eight, twenty seven, sixty four, one hundred and twenty five, and one thousand.

2:27Practise reading this table backwards as well as forwards. Check any uncertain root by multiplying its proposed answer the required number of times. An exam question can say, write down the value of √169. Recall that 13 × 13 = 169. Therefore, the square root is thirteen. The command write down usually allows a direct answer, but the multiplication explains why it is correct. Next, work out 5² multiplied by the cube root of eight.

2:585² = 5 × 5, giving twenty five. The cube root of eight is two because 2 × 2 × 2 = 8. The expression becomes 25 × 2. The answer is fifty. Other powers follow the same repeated multiplication idea. 2⁵ contains five factors of two and equals thirty two. A fourth root reverses raising to the fourth power. The fourth root of eighty one is three because four factors of three multiply to eighty one.

3:29Work out the fourth root of 81 + 2⁵. Four factors of three give nine, then twenty seven, then eighty one, so the root is three. Five factors of two give four, eight, sixteen, and finally thirty two. Adding three and thirty two gives thirty five.

Decimals, negatives and calculator answers

3:46Powers and roots also apply to decimals and fractions. 0.5² is 0.5 × 0.5, giving zero point two five. A fraction is cubed by cubing its numerator and denominator separately. So the cube of one half is 1/8. Brackets change the meaning of a power applied to a negative number. Negative three in brackets, squared, gives positive nine because both factors are negative. Without brackets, the expression means the negative of 3², which is negative nine.

4:16Also, 9 + 16 makes twenty five, whose square root is five. Adding the separate roots of nine and sixteen gives 3 + 4, which is seven, so roots do not split over addition. These recall facts and repeated multiplication methods must work without a calculator on Paper two. Paper four allows a calculator, which helps with roots that are not whole numbers. √2 is approximately one point four one to three significant figures.

4:42Unless instructed otherwise, use three significant figures for answers that are not exact, and do not round intermediate results. A positive whole number power tells you how many equal factors to multiply. A root reverses the matching power, so you can check it by powering your answer. Remember that the square root symbol gives the non-negative value. With these connections, the notation describes a calculation you understand, rather than a rule you merely remember.