## What you need to know

### Standard Form

**Standard form** is a shorthand of expressing **VERY LARGE** or **VERY SMALL** numbers. It is often used across all kinds of sciences and therefore understanding this topic will also help with GCSE science revision.

Make sure you are happy with the following topics before continuing:

## 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 a **whole number** between \mathbf{1} and \mathbf{10}, in other words, 1\leq A<10.

2) Standard from is ways \textcolor{blue}{10} to the power something (\textcolor{limegreen}{n})

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

**For example,**

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

### Example 1: Standard Form into Big Numbers

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.

**Three Easy Steps**

**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}} to fill in the space created as the decimal point has move.

**Step 3:** Remove the original **decimal point**.

This gives the answer to be:

4.2\times10^{5}=\textcolor{red}{420000}

### Example 2: Standard Form – Small Numbers

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

**Three Easy Steps**

**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}} to fill in the space created as the decimal point has move.

**Step 3:** Remove the original **decimal point**.

Therefore, we have concluded that 2.8\times10^{-4}=0.00028.

### Example 3: Big Numbers into Standard Form

Write 56,700,000 in standard form.

**Three Easy Steps**

**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 out 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

### Example 4: Writing Small Numbers in Standard Form

Write 0.0000099 in standard form.

**Three Easy Steps**

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

**Step 2: **Count the number of times the decimal point has moved to the **right**, this will become out power (\textcolor{limegreen}{n}), in this case \textcolor{limegreen}{6}

**Step 3:** We have moved to the ** right** meaning it will be \textcolor{limegreen}{-6} not +6

### Example 5: Multiplying Standard Form

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

To do this we must 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}

We must multiple 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}}

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 21=2.1\textcolor{Orange}{\times10}, then we get that

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

### Example 6: Dividing Standard Form

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.

We need to 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}}

\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})

### Take Note

Purely standard form questions are not that common in GCSE exams, you may get a straight forward question, but more likely, questions will involve standard form integrated into other topic questions. In this way, there may be multiple marks across your GCSE papers for being able to use standard form.

**Note:** With standard form, if the power is a positive number, it’s a big number, and if the power is negative, it’s a small number i.e. less than 1.

### Example Questions

1) What is 1.15\times10^{-6} in decimal notation?

The power is negative, so this is going to be a very small number. As the power of ten is -6, we want to divide the number 1.15 by 10 six times, and so we will move the decimal point six places to the left.

1.15\times10^{-6}=0.00000115.

2) What is 5,980,000 in standard form?

In this case, the power of 10 is going to be positive.

So, if we move the decimal point in 5,980,000 to the left six places it becomes 5.98. Therefore, we get that,

5,980,000=5.98\times10^{6}

3) What is 0.0068 in standard form?

By considering the position where the first non-zero digit is compared to the units column we find,

0.0068=6.8\times10^{-3}

as the 6 is 3 places away from the units column.

4) Calculate 5,600,000\div800

Give your answer in standard form.

First, write each of the numbers in standard form i.e. 5.6\times10^6 and 8\times10^2

(5.6\times10^6)\div(8\times10^2)=(5.6\div8)\times(10^6\div10^2)

Using the formula 10^a\div10^b=10^{a-b} we can rewrite the eqaution as,

(5.6\div8)\times10^{6-2}=0.7\times10^4

Standard form requires the number be between 1 and 10, so adjusting by a factor of 10, we have,

0.7\times10^4=7×10^{-1} \times10^4=7\times10^3

5) Calculate (2.5\times10^{4})\times(6\times10^{-9}).

Give your answer in standard form.

We will split up this multiplication, multiplying the initial numbers together and the powers of 10 together separately. Firstly,

2.5\times6=15.

Secondly, using the multiplication law of indices,

10^4\times10^{-9}=10^{4+(-9)}=10^{-5}

So, we get

\begin{aligned}(2.5\times10^{4})\times(6\times10^{-9})&=2.5\times6\times10^4\times10^{-9}\\&=15\times10^{-5}\end{aligned}

Standard form requires the number be between 1 and 10, so adjusting by a factor of 10, we have,

15\times10^{-5}=1.5\times10\times10^{-5}=1.5\times10^{-4}

### Worksheets and Exam Questions

#### (NEW) Standard Form - Exam Style Questions - MME

Level 4-5#### Standard Form (1) - Drill Worksheet

Level 4-5#### Standard Form (2) - Drill Worksheet

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