Powers (Indices) and Roots


Introduction

  • 52 is shorthand for 5 x 5
  • 53 is shorthand for 5 x 5 x 5
  • 54 is shorthand for 5 x 5 x 5 x 5

etc., etc.


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 :

52   :   five to the power two   or      five squared
53   :   five to the power three   or       five cubed

Note : 51 = 5

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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

4 X 4 = 16 cm2 (16 square centimeters)

Quick Quiz     What is the square of the following numbers

  1. 1.    7
  2. 2.    6
  3. 3.    12
  4. 4.    13
  5. 5.    20
  1. 6.    100
  2. 7.   41
  3. 8.    19
  4. 9.    30

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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

4 X 4 X 4 = 64 cm3 (64 cubic centimeters)

Quick Quiz     What is the cube of the following numbers

  1. 1.    7
  2. 2.   6
  3. 3.   12
  4. 4.   13
  5. 5.   20
  1. 6.   100
  2. 7.   41
  3. 8.   19
  4. 9.   30

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Powers of Ten



102 = 10 x 10 = 100

103 = 10 x 10 x 10 = 1 000

104 = 10 x 10 x 10 x 10 = 10 000

105 = 10 x 10 x 10 x 10 x 10 = 100 000

106 = 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000

Note that the power of ten, as shown on the left, is equal to the number of zeroes in the expressions on the right.

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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.

(-3) × (-3) = 9


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.

Quick Quiz     What is the square root of the following numbers

  1. 1.     121
  2. 2.    169
  3. 3.    49
  4. 4.    3
  5. 5.    70

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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.

  • The first part is a number greater than 0 and less than 10
  • The last part is a power of ten


If 1 000 can be written as

103

then 3 000 can be written as

3 x 103

Consider other examples

493 4.93 x 102
2 014 000 2.014 x 10 6
1 964 1.964 x 103

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.

For very large numbers, rounding is very common, e.g. a number like

2.09 x 1015

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.

Quick Quiz     Express the following in standard form notation

  1. 1.    30
  2. 2.   121 965
  3. 3.   12 369
  4. 4.   564
  5. 5.   2 341 421


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

Quick Quiz     Express the following as 'normal' numbers

  1. 1.    3 × 104
  2. 2.   9.321 × 106
  3. 3.   1.3453 × 102
  4. 4.   5.321 × 106
  5. 5.   2.1 × 109

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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?


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