Standard Form

Standard form, or standard index form as it’s sometimes called, is a really useful way of writing really big or really small numbers. It’s also really popular with examiners – so make sure you know it!

What Is Standard Form?

Standard form is a way of writing out huge or tiny numbers by taking a number between 1 and 10 and multiplying it by a power 10. For example, we know that 103 is 1000. Therefore 3,520 an be written as 3.52 x 103

This pretty much explains it all:standard formJust remember:

  • A is always between 1 and 10
  • N shows you how many places the decimal point moves
  • For negative values of n you get a small number, for positive values you get a large number.

How Do You Convert Between Ordinary Numbers And Standard Form?

The easiest way to do this is to look at an example.

Example: Express 45,700 in standard form.

First you write the number that will be between one and ten at the front of your standard form answer. Here that number will be 4.57.

You then see how many places the decimal point would have to move from this ordinary number to get to this number.You have to move the decimal point 4 places to change 45700.00 to become 4.57.

So your answer is:

4.57 x 104

Example: Express 3.2 x 10-4 as an ordinary number

To do this you just have to look at what the power of ten is in the question and move the decimal point this many places. If it’s a negative power you move the decimal point moves left, it it’s positive it moves right.

Here you move the decimal place 4 places left. So the answer is:


Multiplying & Dividing Numbers In Standard Index Form

This is fairly easy. All you need to do is:

  • Group the front numbers and powers on ten together
  • Multiply or divide the front numbers and the powers of ten separately (you may want to look at the Powers, Roots and Indices revision note)
  • Make sure your answer is still in standard form

Example: What is 340 000 ÷ (6.4 x 109) ? Express your answer in standard form.

(3.4 x 105) ÷ (6.4 x 109)

Split the front numbers and powers of ten

(3.4 / 6.4) x (105 / 109)

= 0.53125 x 10-4

Express in standard form

= 5.3125 x 10-5

Adding & Subtracting Numbers In Standard Index Form

Here’s how you do this:

  • Get the powers of ten to be the same
  • Add or subtract the front numbers
  • Convert to standard form

Example: (4.5 × 104) + (6.45 × 105)

Get powers of ten to be the same.

(4.5 × 104) + (64.5 × 104)

Add the front numbers.

(4.5 + 64.5) x 104

= 69 x 104

Express in standard form

= 6.9 x 105

example questionsHere’s a bunch of example questions. To reveal the answer click on the “answer” tab.

Express 4733 in standard form
4.733 x 103
Express 0.00000081 in standard form
8.1 x 10-7
(9 x 106) x (6 x 108)
5.4 x 1015
(2.45 × 10-9) + (4.93 × 10-6)
This one is difficult.

Remember first you get the powers of ten to be the same:

(0.00245 × 10-6) + (4.93 × 10-6)

You then just need to add together the front numbers

4.93245 × 10-6

As the front number is between 1 and 10, the answer is already in standard form.