Scientific notation calculations using index laws

Lesson

We've already learnt about how to use standard form to write really big or really small numbers. Remember they are written in the form $a\times10^b$a×10b. Since these numbers are all written in relation to a power of 10, numbers written in standard form always have indices (or powers). As such, we can simplify expressions written in standard form using index laws, such as the multiplication law or the division law.

If you are adding or subtracting numbers written in standard form, you need to make sure that the powers of ten are the same. To do this, you may need to factorise some of the powers of 10 based on the rules of the multiplication law. Then you can use the decimal number, rather than the number standard form and solve the problem. The process goes:

 $a\times10^n\pm b\times10^m$a×10n±b×10m $=$= $a\times10^n\pm b\times10^{m-n}\times10^n$a×10n±b×10m−n×10n $=$= $\left(a\pm b\times10^{m-n}\right)\times10^n$(a±b×10m−n)×10n

Tip

Make sure you check that your final answer is expressed appropriately in standard form.

Examples

Question 1

Use index laws to simplify $2\times10^6\times6\times10^5$2×106×6×105. Give your answer in standard form.

Think: Let's simplify the expression first. Remember to express our answer in standard form, we'll need to express the coefficient as a value between $1$1 and $10$10 and then multiply it by the correct power of $10$10.

Do:

 $2\times10^6\times6\times10^5$2×106×6×105 $=$= $12\times10^{6+5}$12×106+5 $=$= $12\times10^{11}$12×1011

$12$12 can be as expressed as $1.2\times10^1$1.2×101. We will use this to write our answer in standard form.

 $1.2\times10^1\times10^{11}$1.2×101×1011 $=$= $1.2\times10^{1+11}$1.2×101+11 $=$= $1.2\times10^{12}$1.2×1012

Question 2

Use index laws to simplify $\frac{4\times10^{-5}}{16\times10^4}$4×10516×104. Give your answer in standard form.

Question 3

Evaluate $4.28\times10^6+3.34\times10^7$4.28×106+3.34×107.