What is standard form?

We always write a number in standard form exactly like this:

\textcolor{red}{A}\times\textcolor{blue}{10}^\textcolor{limegreen}{n}

The Three Key Rules

1)   \textcolor{red}{A} must be any between \mathbf{1} and \mathbf{10}, in other words, 1\leq A<10.

2)   Standard form is always \textcolor{blue}{10} to the power of something (\textcolor{limegreen}{n})

3)   \textcolor{limegreen}{n} must be a whole number, this is the number of places the decimal point moves.

e.g. \textcolor{red}{4.2}\times\textcolor{blue}{10}^\textcolor{limegreen}{5}

Skill 1: Standard Form into Large Numbers

Example: Express 4.2\times\textcolor{limegreen}{10^5} as a number not in standard form.

Firstly, recall that \textcolor{limegreen}{10^5}=\textcolor{limegreen}{10}\times\textcolor{limegreen}{10}\times\textcolor{limegreen}{10}\times\textcolor{limegreen}{10}\times\textcolor{limegreen}{10}.

Then, we get  4.2\times10^5=4.2\times10\times10\times10\times10\times10

Multiplying by 10 means moving the decimal place to the right, here we must do it 5 times:

Step 1: Write 4\textcolor{red}{.}2 out and move the decimal place \textcolor{red}{5} jumps to the right.

Step 2: Add \textcolor{limegreen}{\mathbf{0}}‘s to fill in the spaces created as the decimal point has moved.

Step 3: Remove the original decimal point.

This gives the answer to be:

4.2\times10^{5}=420000

Skill 2: Standard Form into Small Numbers

Example: Write 2.8\times\textcolor{limegreen}{10^{-4}} in decimal notation.

This is different because the power is negative, but it’s actually no harder. We know,

\textcolor{limegreen}{10^{-4}}=\textcolor{limegreen}{\dfrac{1}{10^4}}

This means we are \textcolor{limegreen}{\text{dividing by }10} \textcolor{limegreen}{\text{four times}} which means we move the decimal point 4 spaces to the left.

Step 1: Write 2\textcolor{red}{.}8 out and move the decimal place \textcolor{red}{4} jumps to the left.

Step 2: Add \textcolor{limegreen}{\mathbf{0}}‘s to fill in the space created as the decimal point has moved.

Step 3: Remove the original decimal point.

Therefore, we have concluded that

2.8\times10^{-4}= 0.00028

Skill 3: Writing Large Numbers in Standard Form

Example: Write 56,700,000 in standard form.

Step 1: Move the decimal point to the left until the number becomes 5.67 (1\leq A<10)

Step 2: Count the number of times the decimal point has moved to the left, this will become our power (\textcolor{limegreen}{n}), in this case \textcolor{limegreen}{7}.

Step 3: We have moved to the left meaning it will be \textcolor{limegreen}{+7} not -7

So,

56,700,000 = 5.67\times10^{\textcolor{limegreen}{7}}

Skill 4: Writing Small Numbers in Standard Form

Example: Write 0.0000099 in standard form.

Step 1: Move the decimal point to the right until the number becomes 9.9 (1\leq A<10)

Skill 5: Multiplying Standard Form

Example: Find the standard form value of (3\times10^8)\times(7\times10^4), without using a calculator.

Step 1: Change the order around of the things being multiplied.

(\textcolor{red}{3}\times\textcolor{limegreen}{10^8})\times(\textcolor{red}{7}\times\textcolor{limegreen}{10^4})=\textcolor{red}{3}\times\textcolor{red}{7}\times\textcolor{limegreen}{10^8}\times\textcolor{limegreen}{10^4}

Step 2: Multiply the numbers and the powers out separately.

(\textcolor{red}{3}\times\textcolor{red}{7})\times(\textcolor{limegreen}{10^8}\times\textcolor{limegreen}{10^4})=\textcolor{red}{21}\times\textcolor{limegreen}{10^{12}}

Step 3: Convert the number at the front to standard form if necessary (1 \leq A < 10).

This answer is not in standard form (21 is not between 1 and 10), and we need it to be. Fortunately, if we recognise that \textcolor{red}{21}=\textcolor{red}{2.1}\textcolor{limegreen}{\times10}, then we get that

\textcolor{red}{21}\textcolor{limegreen}{\times10^{12}}=\textcolor{red}{2.1}\textcolor{limegreen}{\times10}\textcolor{limegreen}{\times10^{12}}=\textcolor{red}{2.1}\textcolor{limegreen}{\times10^{13}}

Skill 6: Dividing Standard Form

Example: Find the standard form value of (\textcolor{red}{8}\times\textcolor{limegreen}{10^{-5}})\div(\textcolor{red}{2}\times\textcolor{limegreen}{10^6}), without using a calculator.

Step 1: Break up the problem and change the order of how we divide things.

\dfrac{\textcolor{red}{8}\times\textcolor{limegreen}{10^{-5}}}{\textcolor{red}{2}\times\textcolor{limegreen}{10^6}}=\dfrac{\textcolor{red}{8}}{\textcolor{red}{2}}\times\dfrac{\textcolor{limegreen}{10^{-5}}}{\textcolor{limegreen}{10^6}}=\textcolor{red}{4}\times\textcolor{limegreen}{10^{-11}}

Step 2: Convert the number at the front to standard form if necessary (1 \leq A < 10).

\textcolor{red}{4} is between 1 and 10, so this answer is in standard form, and so we are done.

(Remember: 10^{-5} - 10^6 = 10^{-5-6} = 10^{-11})