 Standard Form Questions | Worksheets and Revision | MME

# Standard Form Questions, Worksheets and Revision

Level 4-5

## Standard Form

Standard form is a shorthand way of expressing VERY LARGE or VERY SMALL numbers. There are 6 key skills that you need to learn.

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

Level 4-5

## 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:

$4.2\times10^{5}=420000$

Level 4-5

## 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

$2.8\times10^{-4}= 0.00028$

Level 4-5

## 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,

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

Level 4-5

## 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 $9.9$ $(1\leq A<10)$ Step 2: Count the number of times the decimal point has moved to the right, this will become our 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 $+6$

So,

$0.0000099 = 9.9\times10^{\textcolor{limegreen}{-6}}$

Level 4-5

## 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

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

Level 4-5

## 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: $10^{-5} - 10^6 = 10^{-5-6} = 10^{-11}$)

Level 4-5

### Example Questions

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.

$1.15\times10^{-6}=0.00000115$.

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,

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

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.

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,

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

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

$(2.5\times10^{4})\times(6\times10^{-9}) =2.5\times6\times10^4\times10^{-9}\\ =15\times10^{-5}$

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 New Official MME

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