Powers (Indices) and Roots
Introduction
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The usual way of describing expressions like this is :
"5 to the power 3",
"5 to the power 4", and so on.
The first two in the list have special 'names' which could also be used as well :
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: 51 = 5
Square Numbers (Squares, for short)
Proceeding by example :- the following are square numbers
| 9 | because 9 = 3 X 3 | therefore 32 (three squared) = 9 |
| 16 | because 16 = 4 X 4 | therefore 42 (four squared) = 16 |
| 25 | because 25 = 5 X 5 | therefore 52 (five squared) = 25 |
| 144 | because 144 = 12 X 12 | therefore 122 (twelve squared) = 144 |
To illustrate the use of the name square : a square with sides of length 4 cms., will have an area of
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Cube Numbers (Cubes, for short)
Taking the above ideas one step further :- the following are cube numbers
| 8 | because 8 = 2 X 2 X 2 | therefore 23 (two cubed) = 8 |
| 27 | because 27 = 3 X 3 X 3 | therefore 33 (three cubed) = 27 |
| 64 | because 64 = 4 X 4 X 4 | therefore 43 (four cubed) = 64 |
| 1728 | because 1728 = 12 X 12 X 12 | therefore 123 (twelve cubed) = 1728 |
To illustrate the use of the name cube : a cube with sides of length 4 cms., will have an area of
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Powers of Ten
Note that the power of ten, as shown on the left, is equal to the number of zeroes in the expressions on the right.
Square Roots
Proceed by example
\[ \sqrt{9} = 3 \]
because 3 x 3 = 9
This is using the inverse procedure to that described in the section on square numbers. You can appreciate that not all integers have integer square roots - in fact, integer square roots are the exception.
The following numbers all have integer square roots : 25, 36, 144.
The following do not have integer square roots : 2, 7, 11.
For example, the square root of 2 is 1.414, to 3 decimal places. You will have to use your calculator for these more complicated roots.
But do not use your calculator for
the square roots of the lower integers - these should be memorized.
You should remember from your previous work that two minus numbers
multiplied together produce a positive number, e.g.
so you can see that actually -3 is also a square root of 9.
In fact, every number has two square roots. The 'additional' one is just the negative of the other.
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Standard Form (Scientific Form)
Standard forms are especially useful for describing large numbers in a compact way.
The standard form consists of two parts multiplied together.
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If 1 000 can be written as
then 3 000 can be written as
Consider other examples
You can see the pattern - shift the decimal point to the left a certain number of
places (leaving just one number to the left of the decimal point), and multiply
by 10 raised to a power equal to the number of places that
the decimal point was moved.
493 4.93 x 102
2 014 000 2.014 x 10 6
1 964 1.964 x 103
has probably been rounded to 2 decimal places. Therefore if this figure was used in further calculations, you would have to make sure your derived answers were not stated to too many figures. The number of significant figures should not be greater than the number of significant figures in the original data used for the calculation.
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To convert numbers written in standard form back into 'normal' numbers, then move the decimal point to the right by the same number of places as the index of 10.
Example
5.345 × 102 = 534.5
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Past Exam Questions
1. Lawrence saves some images onto his floppy disk. Each image requires 35 000 bytes of memory. How many whole images can he save if the floppy disk has a memory equivalent to 1.2 × 106 bytes?