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Academic Skills Advice
Indices Refresher Sheet 1
What do Powers/Indices mean?:
𝑥2
is said: ‘ 𝑥 squared ‘ and means 𝑥 × 𝑥
𝑥3
is said: ‘𝑥 cubed’ and means 𝑥 × 𝑥 × 𝑥
𝑥 4 is said: ‘𝑥 to the power of 4’ and means 𝑥 × 𝑥 × 𝑥 × 𝑥
etc….
The superscript numbers (2, 3 & 4 above) are known as indices or powers. For all other
powers (apart from 2 & 3) you would say 𝑥 to the power of.
Examples:
32 = 3 × 3 = 9
53 = 5 × 5 × 5 = 125
34 = 3 × 3 × 3 × 3 = 81
25 = 2 × 2 × 2 × 2 × 2 = 32
(3 squared)
(5 cubed)
(3 to the power of 4)
(2 to the power of 5)
What do roots mean?:
Roots are the opposite of powers.

√𝟒 = 2 or -2 (because 2 x 2 and -2 x -2 = 4)
We know that 2 x 2 = 22 = 4, so reversing it gives: √4 = 2 (we say square root of 4)
Also notice: -2 x -2 = (-2)2 = 4 so also √4 = -2

𝟑
√𝟐𝟕 = 3 (because 3 x 3 x 3 = 27)
3
We know: 3 x 3 x 3 = 33 = 27, therefore √27 = 3 (we say cubed root of 27).
Also notice: -3 x -3 x -3 = (-3)3 = -27 so there is only 1 root for a cubed number.

𝟒
√𝟏𝟔 = 2 or -2 (because 2 x 2 x 2 x 2 = 16 and -2 x -2 x -2 x -2 = 16)
4
We know that 2 x 2 x 2 x 2 = 24 = 16, so reversing gives: √16 = 2 (we say 4th root of 16)
4
Also notice: -2 x -2 x -2 x -2 = (-2)4 = 16 so also √16 = -2
Remember:
√(𝑥 2 ) = 𝑥
2
(√𝑥) = 𝑥
(The square root of 𝒙𝟐 takes you back to 𝑥 as they cancel each other out)
(if you square √𝑥 you’re back to 𝑥 as they cancel each other out)
2
(√16) = (4)2 = 16
√(32 ) = √9 = 3
The general formula is:
𝑛
√𝑥 𝑛 = 𝑥
1
H Jackson 2008 / Academic Skills
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Practice Questions:
THEREFORE
1) Complete all the blank spaces in the table:
No.
Meaning
Answer
2
1
1x1
1
1
2
2
2x2
4
4
2
3
3x3
9
9
42
52
62
72
82
92
102
Answer/s
+1 or -1
+2 or –2
+3 or –3
Reason
1x1=1 AND -1x-1=1
2x2=4 AND -2x-2=4
3x3=9 AND -3x-3=9
13
23
1x1x1
2x2x2
1
8
1
3
8
1
2
1X1X1 = 1
2X2X2 = 8
33
3x3x3
27
3
3
3X3X3 = 27
THEREFORE
4
3
3
53
63
73
83
9
27
3
3
3
3
3
3
3
103
3
2) What does a number squared mean? (eg 72 )
3) What does the “square root of a number” mean? (eg √9) – what exactly are you finding?
4) Find the square root of 5 on your calculator. What would you expect to get if you
squared this answer?
5) Find the following (you may use a calculator if necessary):
25
109
4
√16
H Jackson 2008 / Academic Skills
36
35
3
√8
44
64
6
√1000000
28
4
√81
9
√512
57
5
√243
7
√78125
2