Standard Forms

For many subjects you will need to convert measurements to other forms. One main form is standard form.

This is normally asked for you to do when a really large or really small number is too difficult to write out fully.

Converting to standard form:

The general form of standard form is A × 10 n . You must take the first few digits of the number and write it as a decimal of a number smaller than 10 but larger than 1. Then write a (x 10) number next to it and have the power of the 10 as many times as it needs to increase/decrease by digits.

Example : Write 26 000 000 in standard index form.

Solution:

26 000 000 = 2.6 × 10 000 000

This can be rewritten as:

2.6 × 10 × 10 × 10 × 10 × 10 × 10 × 10

= 2.6 × 10 7

Another example: 0.000547 in standard form

= 5.47 × 0.0001

= 5.47 × 10 -4

To convert a standard form number into its real/long number simply do the reverse

Adding/subtracting standard form numbers:

You must convert the number into the ordinary number, then do the calculations, then reconvert back into standard form. (Or if you really like standard form just make sure each number is to the same power of 10 then you do the maths!)

Example:

4.5 × 104 + 6.45 × 105

= 45,000 + 645,000

= 690,000

= 6.9 × 105

Multiplying/dividing standard form numbers:

Now you just do what you normally do with each of the numbers. You must remember the indices rules that:

  • To multiply powers you add, eg, 105 × 103 = 108
  • To divide powers you subtract, eg, 105 ÷ 103 = 102

 

E.g.:   Simplify (2 × 103) × (3 × 106)

Multiply 2 by 3 and add the powers of 10:

2 x 3 = 6

103 + 106 = 109

(2 × 103) × (3 × 106) = 6 × 109

 

 

Try this next one and write your answer in the comments:

Simplify (36 × 105) ÷ (6 × 103)

 

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