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Adding and subtracting decimals

Multiplying and dividing decimals

Laws of indices Negative numbersPlace value

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Here we will learn about **adding and subtracting numbers in standard form**.

There are also adding and subtracting numbers in standard form* *worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Adding and subtracting in standard form **works in a similar way to adding and subtracting ordinary numbers. There are two methods we can use.

We can** **either convert standard form to ordinary numbers then use the column method for addition or subtraction, or we can adjust the numbers so that they have the same power of ten and then use addition or subtraction.

E.g.

(4\times10^{3})+(6\times10^{2})

Converting to ordinary numbers first: 4000 + 600 = 4600

However, this method is not very efficient especially for very large and very small numbers.

To add and subtract numbers in standard we can first convert the numbers so that they have the** same power of ten**.

E.g.

Using standard form: (4\times10^{3})+(0.6\times10^{3})=(4.6\times10^{3})

In order to add and subtract numbers in standard form:

**Convert one of the numbers so that both numbers have the same power of ten. Select the number with the lower power of 10.****Add/subtract the decimals.****Write your answer in standard form.**

Get your free adding and subtracting in standard form worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free adding and subtracting in standard form worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONWork out:

\[ (5\times10^{5})\quad+\quad(2\times10^{4})\]

**Convert one of the numbers so that both numbers have the same power of ten. Select the number with the lower power of 10.**

10^{4} is the lowest power of ten. Multiply it by 10 so it also becomes 10^{5}

10^{4} x 1= 10^{5}

Divide

2** Add & subtract the decimals. **

3** Write your answer in standard form. **

( 5 x 10^{5}) + ( 0.2 x 10^{5}) = 5.2 x 10^{5}

Work out:

\[(7\times10^{-3})\quad+\quad(6\times10^{-4})\]

10^{-4} is the lowest power of ten. Multiply it by 10 so it also becomes 10^{-3}

\[10^{-4}\times10\quad=\quad10^{-3}\]

Divide 6 by 10 to maintain the value.

\[6\div10 = 0.6\]

\[6\times10^{-4}\quad=\quad0.6\times10^{-3}\]

** Add/subtract the decimals. **

\[7 + 0.6 = 7.6\]

**Write your answer in standard form.**

\[(7\times10^{-3})\quad+\quad(0.6\times10^{-3})\quad=\quad7.6\times10^{-3}\]

Work out

\[(8.1\times10^{7})\quad+\quad(2.5\times10^{5})\]

10^{5} is the lowest power of ten. Multiply it by 100 so it also becomes 10^{7}

\[10^{5}\times100\quad=\quad10^{7}\]

Divide 2.5 by 100 to maintain the value.

\[2.5\div100 = 0.025\]

\[2.5\times10^{5}\quad=\quad0.025\times10^{7}\]

**Add/subtract the decimals.**

Write 8.1 + 0.025 = 8.125

**Write your answer in standard form.**

\[(8.1\times10^{7})\quad+\quad(0.025\times10^{7})\quad=\quad8.125\times10^{7}\]

Calculate

\[(8\times10^{4})\quad-\quad(6\times10^{3})\]

Write your answer in standard form.

10^{3} is the lowest power of ten. Multiply it by 10 so it also becomes 10^{4}

\[10^{3}\times10\quad=\quad10^{4}\]

Divide 6 by 10 to maintain the value.

\[6\div10 = 0.6\]

\[6\times10^{3}\quad=\quad0.6\times10^{4}\]

**Add/subtract the decimals.**

\[8 – 0.6 = 7.4\]

**Write your answer in standard form.**

\[(8\times10^{4})\quad+\quad(0.6\times10^{4})\quad=\quad7.4\times10^{4}\]

Calculate

\[(8\times10^{-2})\quad-\quad(3\times10^{-3})\]

Write your answer in standard form.

10^{-3} is the lowest power of ten. Multiply it by 10 so it also becomes 10^{-2}

\[10^{-3}\times10\quad=\quad10^{-2}\]

Divide 3 by 10 to maintain the value.

\[3\div10 = 0.3\]

\[3\times10^{-3}\quad=\quad0.3\times10^{-2}\]

**Add/subtract the decimals.**

\[8 – 0.3 = 7.7\]

**Write your answer in standard form.**

\[(8\times10^{-2})\quad-\quad(0.3\times10^{-2})\quad=\quad7.7\times10^{-2}\]

Calculate

\[(6.2\times10^{4})\quad-\quad(1.8\times10^{2})\]

Write your answer in standard form.

10^{2} is the lowest power of ten. Multiply it by 100 so it also becomes 10^{4}

\[10^{2}\times100\quad=\quad10^{3}\]

Divide 1.8 by 100 to maintain the value.

\[1.8\div100 = 0.018\]

\[1.8\times10^{2}\quad=\quad0.018\times10^{4}\]

**Add/subtract the decimals.**

\[6.2 – 0.018 = 6.182\]

**Write your answer in standard form.**

\[(6.2\times10^{4})\quad-\quad(0.018\times10^{4})\quad=\quad6.182\times10^{4}\]

**Using the column method**

While this is not wrong, it is an inefficient method and can lead to errors when converting the numbers, especially with very large and very small numbers with many place holders.

E.g.

Work out (8\times10^{4})\quad+\quad(6\times10^{3})

\[8000 + 600 = 8600 = 8.6\times10^{4}\]\]

Using standard form:

\[(8\times10^{4})\quad+\quad(0.6\times10^{4})\quad=\quad8.6\times10^{4}\]

**Not converting solutions to standard form**

After calculating with standard form, a common mistake is not ensuring the number is in standard form.

Remember to be in standard form the number needs to have two parts, the first part should between 1 and 10 (

E.g.

62\times10^{7} is not in standard form as

In standard for this should be written as 6.2\times10^{8}

**Negative powers**

A common mistake is to become mixed up when using negative powers.

E.g.

10^{-3} is smaller than 10^{-2} because -3 is less than -2 .

10^{-3}=0.001 and 10^{-2}=0.01 so 10^{-2} is greater than 10^{-3}

1. Work out (2\times10^{6})\quad+\quad(3\times10^{5}) . Write your answer in standard form

23 \times 10^{5}

5\times 10^{11}

3.2 \times 10^{5}

2.3 \times 10^{6}

3 \times 10^{5}=0.3 \times 10^{6}\\
(2 \times 10^{6})+(0.3 \times 10^{6}=2.3 \times 10^{6}

2. Work out (4\times10^{-5})\quad+\quad(7\times10^{-6}) . Write your answer in standard form.

4.7 \times 10^{-5}

4.7 \times 10^{-4}

7.4 \times 10^{-4}

47 \times 10^{-6}

7 \times 10^{-6}=0.7 \times 10^{-5}\\
(4 \times 10^{-5}) + (0.7 \times 10^{-5})=4.7 \times 10^{-5}

3. Work out (6.4\times10^{8})\quad+\quad(3.5\times10^{6}) . Write your answer in standard form.

6.75 \times 10^{8}

643.5 \times 10^{6}

6.435 \times 10^{8}

9.9 \times 10^{14}

3.5 \times 10^{6} = 0.035 \times 10^{8}\\
(6.4 \times 10^{8}) + (0.035 \times 10^{8}) = 6.435 \times 10^{8}

4. Work out (7\times10^{4})\quad-\quad(4\times10^{3}) . Write your answer in standard form.

7.4 \times 10^{4}

66 \times 10^{3}

6.6 \times 10^{4}

3 \times 10^{4}

4 \times 10^{3} = 0.4 \times 10^{4}\\
(7 \times 10^{4})-(0.4 \times 10^{4})=6.6 \times 10^{4}

5. Work out (5\times10^{-3})\quad-\quad(2\times10^{-4}) . Write your answer in standard form

4.8 \times 10^{-3}

1.5 \times 10^{-3}

4.8 \times 10^{-4}

3 \times 10^{1}

2 \times 10^{-4} = 0.2 \times 10^{-3}\\
(5 \times 10^{-3})-(0.2 \times 10^{-3}) = 4.8 \times 10^{-3}

6. Work out (7.8\times10^{5})\quad-\quad(2.3\times10^{3}) . Write your answer in standard form.

7.57 \times 10^{5}

7.823 \times 10^{5}

777.7 \times 10^{3}

7.777 \times 10^{5}

2.3 \times 10^{3} = 0.023 \times 10^{5}\\
(7.8 \times 10^{5})-(0.023 \times 10^{5}) = 7.777 \times 10^{5}

1. The table below shows the population of several cities

City |
Population |

London | 8.9 x 106 |

Manchester | 5.5 x 105 |

Birmingham | 1.1 x 106 |

Oxford | 1.1 x 105 |

Work out the total population of London and Manchester. Give your answer in standard form.

**(3 marks)**

Show answer

Simplifying by writing the two populations as an ordinary number:

London: 8900000

Manchester: 550000

OR

converting 5.5\times10^{5} to 0.55\times10^{6}

**(1)**

Adding the numbers:

8900000 + 550000 = 9450000

OR

8.9\times10^{6} +0.55\times10^{6} = 9.45 \times 10^{6}**(1)**

**(1)**

2. Work out 7\times10^{6}\quad-\quad4\times10^{5} .

Give your answer in standard form.

**(3 marks)**

Show answer

Simplifying by writing the two numbers as an ordinary numbers:

7000000 – 400000

OR

converting 4\times10^{5} to 0.4\times10^{6}

**(1)**

Subtracting the numbers:

7000000 – 400000 = 6600000

OR

7\times10^{6} – 0.4\times10^{6} = 6.6 \times 10^{6}**(1)**

**(1)**

3. Show that

2.4\times10^{2}\quad+\quad3.7\times10^{3}\quad=\quad3.94\times10^{3}

**(2 marks)**

Show answer

240 + 3700 or 0.24\times10^{3} or 39.4\times10^{2}

**(1)**

Correct working shown

**(1)**

You have now learned how to:

- Add numbers in standard form
- Subtract numbers in standard form

- Convert between ordinary numbers and standard form
- Multiply numbers in standard form
- Divide numbers in standard form
- Linear equations
- Quadratic equations
- Surds

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