Calculating standard form without a calculator

Adding and subtracting

When adding and subtracting standard form numbers, you have to:

  1. convert the numbers from standard form into ordinary numbers
  2. complete the calculation
  3. convert the number back into standard form

Example

Calculate (4.5 \times 10^4) + (6.45 \times 10^6).

= 45,000 + 6,450,000

= 6,495,000

= 6.495 \times 10^6

Multiplying and dividing

When multiplying and dividing you can use index laws:

  1. multiply or divide the first part of the numbers
  2. apply the index laws to the powers of 10
  3. check whether the first part of the number is between 1 and 10
curriculum-key-fact
To multiply powers of 10, add the powers together, eg 106 x 104 = 106 + 4 = 1010. To divide powers of 10, subtract the powers, eg 107 ÷ 102 = 107 - 2 = 105.

Example 1

Work out (3 \times 10^3) \times (3 \times 10^9).

Multiply the first numbers – which in this case is 3 \times 3 = 9.

Apply the index law to the powers of 10:

  • 10^3 \times 10^9 = 10^{3 + 9} = 10^{12}
  • (3 \times 10^3) \times (3 \times 10^9) = 9 \times 10^{12}

As 9 is between 1 and 10, this number is in standard form.

Example 2

Work out (4 \times 10^9) \times (7 \times 10^{-3}).

Multiply the first numbers 4 \times 7 = 28.

Apply the index law to the powers of 10

  • 10^9 \times 10^{-3} = 10^{9 + -3} = 10^6
  • (4 \times 10^9) \times (7 \times 10^{-3}) = 28 \times 10^6

28 is not between 1 and 10, so 28 \times 10^6 is not in standard form. To convert this to standard form, divide 28 by 10 so that it is a number between 1 and 10. To balance out this out, multiply the second part by 10 which gives 107.

28 \times 10^6 and 28 \times 10^7 are equivalent but only 2.8 \times 10^7 is written in standard form.

So: (4 \times 10^9) \times (7 \times 10^{-3}) = 2.8 \times 10^7

Question

Calculate (2 \times 10^7) \div (8 \times 10^2).

  1. Divide the first part of the numbers: 2 \div 8 = 0.25
  2. Apply the index laws to the powers of 10
  3. Check whether the first part of the number is between 1 and 10
  4. 10^7 \div 10^2 = 10^{7 - 2} = 10^5
  5. So: (2 \times 10^7) \div (8 \times 10^2) = 0.25 \times 10^5

0.25 is not between 1 and 10, so 0.25 \times 10^5 is not in standard form. To convert this to standard form, multiply 0.25 by 10 so that it is a number between 1 and 10. To balance out that multiplication of 10, divide the second part by 10 which gives 104. So 0.25 \times 10^5 and 2.5 \times 10^4 are equivalent but only the second is written in standard form.

So: (2 \times 10^7) \div (8 \times 10^2) = 2.5 \times 10^4