Calculations in standard index form

Multiplying and dividing numbers in standard form

Here you can use the rules for multiplying and dividing powers. Remember these rules:

  • To multiply powers you add, eg {10}^{5} \times {10}^{3} = {10}^{8}
  • To divide powers you subtract, eg {10}^{5} \div {10}^{3} = {10}^{2}

Example

Simplify (2 \times {10}^{3}) \times (3 \times {10}^{6}).

Solution

It is useful to separate the numbers and the indices, and then multiply the {2} by {3} and add the powers of {10}:

(2\times{10}^{3})\times(3\times{10}^{6})=2\times3\times{10}^{3}\times{10}^{6}

={6}\times{10}^{9}

A similar method is used when dividing:

Example

Simplify (6\times10^4)\div(2\times10^2).

Solution

Another way to write a division is to write it in the form of a fraction. The numbers and the indices can then be separated, before dividing the {6} by {2} and subtracting the powers of {10}:

(6\times10^6)\div(2\times10^2)=\frac{(6\times10^6)}{(2\times10^2)}

=\frac{6}{2}\times\frac{{10}^{5}}{{10}^{3}}={3}\times{10}^{2}.

Question

Simplify (8.4\times10^5)\div(4\times10^3).

Did you get 2.1\times{10}^{2}?

Remember to write the division in the form of a fraction to start.

(8.4\times10^5)\div(4\times10^3)=\frac{(8.4\times10^5)}{(4\times10^3)}

Remember you should first work out {8.4}\div{4}, then subtract the powers of {10} (because it is division), like this:

\frac{(8.4\times10^5)}{(4\times10^3)}=\frac{8.4}{4}\times\frac{10^5}{10^3}=2.1\times{10}^2.

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