1.4 Powers and roots Key words
Learning objective • To understand and use powers and roots
cube
cube root
power
square
square root
Squares, cubes and powers Look at these 3D shapes. Is shape B twice as big, four times as big or eight times as big as shape A? How many times bigger is shape C than shape A? You calculate the area of each face of shape C by squaring the side length. 3 × 3 or 32 = 9 You say this as ‘three squared’.
A
B
C
You calculate the volume of the shape by cubing the side length. 3 × 3 × 3 or 33 = 27 You say this as ‘three cubed’. The small digits 2 and 3 are called powers. The power shows how many lots of the number are being multiplied together. A power can be of any size. For example: 34 is equal to 3 × 3 × 3 × 3 = 9 × 9 = 81 and you say it as ‘three to the power 4’.
•
Example 9 Calculate the value of each number. a
53
b
7.52
c
(-2)4
a 53 = 5 × 5 × 5 = 125 b 7.52 = 7.5 × 7.5 = 56.25 You could also do this on a calculator. Most calculators have a button for squaring, usually marked x 2 . They also have a button for cubing, usually marked x 3 . c (-2)4 = −2 × −2 × −2 × −2 = 4 × 4 = 16
Square roots and cube roots To find the area of a square you ‘square’ the side. The inverse process, to work out the side length of a square from its area, is finding the square root. If you know that the area of a square is 25 cm2 then the side length will be the square root of 25, written as 25. This is the number that will give 25, when you multiply it by itself.
25 = 5, because 5 × 5 = 25 If you know that the volume of a cube is 64 cm3, and you want to know its side length, you need to find the cube root. This will be the number that, when multiplied by itself and then multiplied by itself again (three ‘lots’ of the number are multiplied together), gives 64. It is written as 3 64 . 3
16
64 = 4, because 4 × 4 × 4 = 64 1 Working with numbers
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