Standard form calculations

In a nutshell

Standard form is used as an easy way to write large or small numbers. There are rules for adding, subtracting, multiplying and dividing numbers that are written in standard form.

Writing a number in standard index form

A number is in standard index form if it is written in the form:

$a\times10^b$

Where the number $a$ is either an integer or a decimal that is always between $1$ and $10$ , and $b$ is a whole number that may be negative or positive.

Write a large or small number in standard form

PROCEDURE

1.	Move the decimal point until the number is between $1$ and $10$ . Count how many spaces the decimal point was moved.
2.	This number between $1$ and $10$ is the value of $a$ .
3.	The number of spaces the decimal point was moved is the value of $b$ . If the decimal point moved to the left, then $b$ is positive. If the decimal point was moved to the right, then $b$ is negative.

Example 1

What is $76000$ in standard index form?

First, write $76000$ as $76000.00$ and move the decimal point until the number is between $1$  and $10$ :

$7\overset{\curvearrowleft \curvearrowleft \curvearrowleft \curvearrowleft}{.6\, 0\,0\,0\,}0\thinspace 0$ , so $a=7.6$ 

The decimal point moved $4$ spaces to the left, so $b=4$ .

$\underline{76000=7.6\times10^4}$

Example 2

What is $0.0000815$ in standard index form?

First, move the decimal point until the number is between $1$ and $10$ :

$0\thinspace \overset{\curvearrowright \curvearrowright \curvearrowright \curvearrowright \curvearrowright}{0\,0\,0\,0\,8.}15$ , so $a=8.15$

The decimal point moved $5$ spaces to the right, so $b=-5$ .

$\underline{0.0000815=8.15\times10^{-5}}$ 

Multiplying and dividing in standard index form

To multiply and divide two numbers in standard form, use the rules of indices. Multiply/divide the numbers as normal, and use rules of indices to multiply or divide the powers of $10$ . To multiply the powers of $10$ , add the powers and to divide powers of $10$ , subtract the powers.

Example 3

What is $(1.5\times10^8)\times(9\times10^{-3})$ ?

Multiply $1.5$ and $9$ :

$1.5 \times 9 = 13.5$

Use rules of indices to multiply the powers of $10$ :

$10^8 \times 10^{-3} =10^{8+(-3)}= 10^5$

Combine the answers to give:

$(1.5\times10^8)\times(9\times10^{-3}) = 13.5 \times 10^5$ 

Adjust the number so that it is written in standard form. $a$ should be between $1$ and $10$ . So change $13.5$ to $1.35$ and increase the power of $10$ by $1$ .

$\underline{(1.5\times10^8)\times(9\times10^{-3}) = 1.35 \times 10^6}$

Example 4

What is $(8\times10^5)\div(1.25\times10^{2})$ ?

Divide $8$ by $1.25$ :

$8 \div 1.25 = 6.4$

Use rules of indices to divide the powers of $10$ :

$10^5 \div 10^{2} = 10^3$

Combine the answers to give:

$\underline{(8\times10^5)\div(1.25\times10^{2}) = 6.4 \times 10^3}$ 

Adding and subtracting in standard index form

To add and subtract numbers in standard index form, make sure that the powers of $10$ are the same first. Then, add/subtract the numbers and adjust the answer so that the number is between $1$ and $10$ .

Example 5

What is $(8\times10^4)+(5\times10^2)$ ?

First, make the powers of $10$ the same. The easiest way to do this is to write:

$8\times10^4=8\times10^2\times10^2=800\times10^2$

Now, add the numbers together:

$(800\times10^2)+(5\times10^2)=(800+5)\times10^2=805\times10^2$

Rewrite the number so that the answer is in standard form:

$805\times10^2=8.05\times10^2\times10^2=8.05\times10^4$

$\underline{(8\times10^4)+(5\times10^2)=8.05\times10^4}$