 What you need to know

Things to remember:

• We can count how many times the decimal place has moved to work out the power.
• When we have a number between 0 and 1, the power will be negative when converting to standard form.

Standard form is typically used to for extremely large or extremely small numbers, such as measuring the speed of light or the size of bacteria. Start form numbers are written like this:

$$a\times10^n$$

Where $a$ is any number between 1 and 10, but not including 10, and $n$ is a positive or negative integer.

Write $65400$ in standard form

We will do this in 4 steps.

Step 1: If you have a whole number, but a decimal point and 0 on the end.

65400.0

Step 2: Put a second decimal point in on the right of the first non-zero number.

6.5400.0

Step 3: Count how many times the first decimal place “moved” and in which direction.

Here, the decimal place moved 4 times to the left.

Step 4: Write in standard form ($a\times10^n$)

• $a$ will be decimal number in Step 2, without the 0s.
• $n$ is the number found in Step 3.
• If the decimal moved left $n$ is positive.
• If the decimal moved right $n$ is negative.

$$65400=6.54\times10^4$$

We can do the same steps for writings numbers less than 1, but this time the power will be negative.

Write $0.00004562$ in standard form

Step 1: If you have a whole number, but a decimal point and 0 on the end.

Our number isn’t a whole number.

Step 2: Put a second decimal point in on the right of the first non-zero number.

0.00004.562

Step 3: Count how many times the first decimal place “moved” and in which direction.

Here, the decimal place moved 5 times to the right.

Step 4: Write in standard form ($a\times10^n$)

• $a$ will be decimal number in Step 2, without the 0s.
• $n$ is the number found in Step 3.
• If the decimal moved left $n$ is positive.
• If the decimal moved right $n$ is negative.

$$0.00004562=4.562\times10^{-5}$$

Example Questions

Step 1: If you have a whole number, but a decimal point and 0 on the end.

Our number isn’t a whole number.

Step 2: Put a second decimal point in on the right of the first non-zero number.

0.003.56

Step 3: Count how many times the first decimal place “moved” and in which direction.

Here, the decimal place moved 3 times to the right.

Step 4: Write in standard form ($a\times10^n$)

$a$ will be decimal number in Step 2, without the 0s.

$n$ is the number found in Step 3.

If the decimal moved left $n$ is positive.

If the decimal moved right $n$ is negative.

$$0.00356=3.56\times10^{-3}$$

Step 1: If you have a whole number, but a decimal point and 0 on the end.

Our number isn’t a whole number.

Step 2: Put a second decimal point in on the right of the first non-zero number.

5.42.023

Step 3: Count how many times the first decimal place “moved” and in which direction.

Here, the decimal place moved 2 times to the left.

Step 4: Write in standard form ($a\times10^n$)

$a$ will be decimal number in Step 2, without the 0s.

$n$ is the number found in Step 3.

If the decimal moved left $n$ is positive.

If the decimal moved right $n$ is negative.

$$542.023=5.42023\times10^{2}$$