Maths
Powers and roots
Squaring, cubing and higher powers are shown by small digits called indices, like 10^{2} and 5^{3}.
The opposite of squaring a number is finding the square root, and the same is true for cubing and cube roots.
This Revision Bite covers:
9 is a square number.
3 × 3 = 9
3 × 3 can also be written as 3^{2}. This is pronounced "3 squared".
8 is a cube number.
2 × 2 × 2 = 8
2 × 2 × 2 can also be written as 2^{3}, which is pronounced "2 cubed".
The notation 3^{2} and 2^{3} is known as index form. The small digit is called the index number or power.
You have already seen that 3^{2} = 3 × 3 = 9, and that 2^{3} = 2 × 2 × 2 = 8.
Similarly, 5^{4} (five to the power of 4) = 5 × 5 × 5 × 5 = 625
and 3^{5} (three to the power of 5) = 3 × 3 × 3 × 3 × 3 = 243.
The index number tells you how many times to multiply the numbers together.
When the index number is two, the number has been 'squared'.
When the index number is three, the number has been 'cubed'.
When the index number is greater than three you say that it is has been multiplied 'to the power of'.
For example:
7^{2} is 'seven squared',
3^{3} is 'three cubed',
3^{7} is 'three to the power of seven',
4^{5} is 'four to the power of five'.
Look at the table and work out the answers. The first has been done for you.
4^{3} |
4 × 4 × 4 | 64 |
2^{7} |
2 × 2 × 2 × 2 × 2 × 2 × 2 | |
7^{2} |
7 × 7 | |
5^{3} |
||
2^{4} |
||
6^{5} |
4^{3} |
4 × 4 × 4 | 64 |
2^{7} |
2 × 2 × 2 × 2 × 2 × 2 × 2 | 128 |
7^{2} |
7 × 7 | 49 |
5^{3} |
5 × 5 × 5 | 125 |
2^{4} |
2 × 2 × 2 × 2 | 16 |
6^{5} |
6 × 6 × 6 × 6 × 6 | 7776 |
All scientific calculators have a 'power' button. This is usually labelled [x^{y}]. This is particularly useful when the index number is large.
To work out 4 ^{10}:
Enter 4
Press the x^{y} button
Enter 10
Press =
You should get the answer 1 048 576.
Use your calculator to find the values of the following:
a) 2^{11}
b) 5^{8}
c) 2^{6} × 3^{5}
a) 2048
b) 390625
c) 2^{6} × 3^{5} = 64 × 243 = 15552
The opposite of squaring a number is called finding the square root.
The square root of 16 is 4 (because 4^{2} = 4 × 4 = 16)
The square root of 25 is 5 (because 5^{2} = 5 × 5 = 25)
The square root of 100 is 10 (because 10^{2} = 10 × 10 = 100)
What is the square root of 4?
2 × 2 = 4, so 2 is the square root of 4.
The symbol '√ ' means square root, so
√ 36 means 'the square root of 36', and
√ 81 means 'the square root of 81'
You will also find a square root key on your calculator.
The opposite of cubing a number is called finding the cube root.
The cube root of 27 is 3 (because 3 × 3 × 3 = 27)
The cube root of 1000 is 10 (because 10 × 10 × 10 = 1000)
What is the cube root of 8?
2 × 2 × 2 = 8, so 2 is the cube root of 8.
How can we work out 2^{3} × 2^{5}?
2^{3} = 2 × 2 × 2
2^{5} = 2 × 2 × 2 × 2 × 2
so 2^{3} × 2^{5} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^{8}
There are 3 twos from 2^{3} and 5 twos from 2^{5}, so altogether there are 8 twos.
In general, 2^{m} × 2^{n} =2^{(m + n)}
2^{5} × 2^{4} = 2^{(5 + 4)} = 2^{9}
2^{7} × 2^{3} = 2^{(7 + 3)} = 2^{10}
The rule also works for other numbers, so
3^{4} × 3^{2} = 3^{(4 + 2)} = 3^{6}
25^{6} × 25^{4} = 25^{(6 + 4)} = 25^{10}
If you divide 2^{5} by 2^{3} you see that some of the 2's cancel:
So 2^{5} ÷ 2^{3} = 2^{2}
In general, 2^{m} ÷ 2^{n} = 2^{(m - n)}
2^{5} ÷ 2^{2} = 2^{(5 - 2)} = 2^{3}
2^{7} ÷ 2^{3} = 2^{(7 - 3)} = 2^{4}
The rule also works for other numbers, so
5^{10} ÷ 5^{3} =5^{(10 - 3)} = 5^{7}
45^{9} ÷ 45^{4} = 45^{(9 - 4)} = 45^{5}
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