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Place value

Multiplying and dividing

Laws of indices Negative numbersThis topic is relevant for:

Here we will learn about **standard form** including how to convert between ordinary numbers and standard form, and how to calculate with numbers in standard form.

There are also 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.

**Standard form** is a way of writing very large or very small numbers by comparing the powers of ten. It is also known as scientific notation.

Numbers in standard form are written in this format:

\[ a\times10^{n}\]

Where

To do this we need to understand the place value of a number

E.g.

Let’s look at the number

\[\begin{aligned}
&10^6 \quad \quad 10^5 \quad \quad 10^4 \quad \quad 10^3 \quad \quad 10^2 \quad \quad 10^1 \quad \quad 10^0 \\
&\; 8 \quad \quad \quad 2 \quad \quad \quad 9 \quad \quad \; \; \; 0 \quad \quad \; \; \; 0 \quad \quad \; \; \; 0 \quad \quad \; \; \; 0
\end{aligned}\]

So **standard form** is

\[ 8.29\times10^{6}\]

Standard form notation is a representation of place value which compliments the decimal number system, as shown in the table below.

Words | Decimal form | Fraction form | Fraction form with powers of ten | Powers of ten | Standard form |

Thousand | 1000 | 1000 | 10^3 | 10^3 | 1\times10^{3} |

Hundred | 100 | 100 | 10^2 | 10^2 | 1\times10^{2} |

Ten | 10 | 10 | 10^1 | 10^1 | 1\times10^{1} |

Unit | 1 | 1 | 10^0 | 10^0 | 1\times10^{0} |

Tenth | 0.1 | \frac{1}{10} | \frac{1}{10^1} | 10^{-1} | 1\times10^{-1} |

Hundredth | 0.01 | \frac{1}{100} | \frac{1}{10^2} | 10^{-2} | 1\times10^{-2} |

Thousand-th | 0.001 | \frac{1}{1000} | \frac{1}{10^3} | 10^{-3} | 1\times10^{-3} |

Any integer or terminating decimal can be written using the standard form notation a\times10^{n}. The table below shows how the value of a can remain the same while the power of ten ‘n’ changes the place value of those digits.

Decimal form | Standard form |

4500 | 4.5\times10^{3} |

450 | 4.5\times10^{2} |

45 | 4.5\times10^{1} |

4.5 | 4.5\times10^{0} |

0.45 | 4.5\times10^{-1} |

0.045 | 4.5\times10^{-2} |

0.0045 | 4.5\times10^{-3} |

Using standard form notation enables us to write very large or very small numbers.

E.g.

67 500 000 000 000 000 000 000 = 6.75\times10^{22} 0.000 000 000 000 037 = 3.7\times10^{-14}Using standard form notation also enables us to compare the size of very large or very small numbers easily.

E.g.

Which is larger, 8560000000000 or 45320000000000?

At a glance it is difficult to tell which is larger, but written in standard form you can compare these numbers very quickly as shown.

Which is larger, 8.56\times10^{12} or 4.532\times10^{13}?

Instantly we can see that 4.532\times10^{13} is the larger number as it has the higher power of ten.

When numbers are written in standard form it can make some calculations neater and quicker to compute. This is particularly the case for calculations involving multiplication, division and powers. However, this is reliant on an understanding of the rules of indices.

Multiplication rule a^{b} \times a^{c} = a^{b+c}

Division rule a^{b} \div a^{c} = a^{b-c}

Zero power rule a^{0} = 1

Power of power rule (a^{b})^{c} = a^{b \times c}

To review this topic see **step-by-step guide**: Laws of indices

E.g.

340000 \times 0.002

In standard form this changes to (3.4 \times 10^{5}) \times (2 \times 10^{-3})

Brackets are often used to show each separate number in standard form but they are not actually necessary 3.4 \times 10^{5} \times 2 \times 10^{-3}

As multiplication is commutative (the order does not matter) we can reorder this calculation to 3.4 \times 2 \times 10^{5} \times 10^{-3}

3.4 \times 2 = 6.8

10^{5} \times 10^{-3} = 10^{5+-3} = 10^{2} (applying the multiplication rule of indices a^{b} \times a^{c} = a^{b+c})

(3.4 \times 10^{5}) \times (2 \times 10^{-3}) = 6.8 \times 10^{2}

340000 \times 0.002 = 680

Let’s look at how to calculate with standard form in more detail.

**Identify the non-zero digits and write these as a decimal number which is greater than or equal to**1 but less than10 **In order to maintain the place value of the number, this decimal number needs to be multiplied by a power of ten****Write the power of ten as an exponent****Write your number in standard form**

E.g.

Convert

- Writing
4500 as a decimal between1 and10 is4.5 4.5 × 1000 10 ^{3}= 1000- So
4500 written in standard form is**4.5 × 10**^{3}

**Convert the power of ten to an ordinary number****Multiply the decimal number by this power of ten****Write your number as an ordinary number**

E.g.

Convert ^{4}

10 ^{4}= 100007.1 × 10000 - So
7.1 × 10 as an ordinary number is^{4}**71000**

**Step-by step guide: **Converting to and from standard form

**Multiply or divide the non-zero numbers****Multiply or divide the powers of ten by adding or subtracting the indices****Write the solution in standard form, checking that the first part of the number is**1 ≤x <10

E.g.

Work out ^{5}) × (2 × 10^{8})

4 × 2 = 8 10 ^{5}× 10^{8}= 10^{13}8 × 10 is in standard form^{13}

E.g.

Work out ^{3}) ÷ (4 × 10^{8})

2 ÷ 4 = 0.5 10 ^{3}÷ 10^{8}= 10^{-5}0.5 × 10 is not in standard form^{-5}

To compensate, you need to divide the power of ten by ^{-6}

So the answer is ^{-6}

** Step-by-step guide:** Multiplying and dividing in standard form

**Convert the number with the lower power of ten so that both numbers have the same power of ten****Add the non-zero integers****Check your answer is in standard form**

E.g. ^{5}) + (6 × 10^{4})

6 × 10 ^{4}= 0.6 × 10^{5}3 + 0.6 = 3.6 3.6 × 10 is in standard form^{5}

E.g. ^{4}) − ( 3 × 10^{3})

3 × 10 ^{3}= 0.3 × 10^{4}9 − 0.3 = 8.7 8.7 × 10 is in standard form^{4}

**Step-by-step guide:** Adding and subtracting in standard form

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

DOWNLOAD FREEGet your free standard form worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEWrite this number in standard form:

52000

**The non-zero digits need to be written as a decimal number**

The number needs to lie between 1≤ x <10

So the number will begin as 5.2…

2**You now need to maintain the value of the number by multiplying that decimal by a power of ten **

\[5.2 \times 10000 = 52000\]

3**Write that power of ten as an exponent**

\[10, 000 = 10^{4}\]

4W**rite your number in standard form**

\[5.2\times10^{4}\]

Write 9.4\times10^5 as an ordinary number.

**Write the exponent as a power of ten**

\[10^5 = 100000\]

**Multiply the decimal number by that power of ten**

\[9.4 \times 100000\]

**Write your answer as an ordinary number**

\[9.4 \times 100000 = 940000\]

Write 0.0068 in standard form.

**The non-zero digits need to be written as a decimal number. The number needs to lie between 1≤ x <10**

So the number will begin as

**Identify what power of ten the decimal needs to be multiplied by in order to preserve place value**

\[6.8 \times \frac{1}{1000} =0.0068\]

**Write that power of ten as an exponent**

\[\frac{1}{1000}=10^{-3}\]

**Write your number in standard form**

\[6.8\times10^{-3}\]

Calculate (5\times10^{4}) \times (7\times10^{8}) .

Write your answer in standard form.

**Multiply the non-zero digits**

\[5 \times 7 = 35\]

**Multiply the powers of ten by adding the powers**

\[10^{4} \times 10^{8} = 10^{12}\]

**Put these two parts together**

\[35\times10^{12}\]

**However, this number is not in standard form as 35 is not a decimal number.**

To convert

To maintain the value of the number, you need to multiply the power of ten by 10, which adds one to the exponent.

\[\begin{aligned}
&\quad \quad \quad \quad 35\times10^{12} \\\\
&\div10 ↓ \quad \quad \quad \quad \quad \quad \times10 ↓\\\\
&\quad \quad \quad \quad 3.5\times10^{13}
\end{aligned}\]

Calculate (8\times10^{7})\div(2\times10^{2}) .

Write your answer in standard form.

**Divide the non-zero digits**

\[8 \div 2 = 4\]

**Divide the powers of ten by subtracting the powers**

\[10^{7} \div 10^{2} = 10^{5}\]

**Put these two parts together**

\[4\times10^{5}\]

**Check your number is in standard form**

This is in standard form as 4 is greater than 1 but less than 10.

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

Write your answer in standard form.

** Convert the number with the smaller power so that both numbers have the same power**

\[2\times10^{3}\]

You need to increase the power, to do this multiply it by

\[\begin{aligned}
&\quad \quad \quad\quad \;\; 2\times10^{3} \\\\
&\div10 ↓ \quad \quad \quad \quad \quad \quad \times10 ↓\\\\
&\quad \quad \quad \quad \;\; 0.2\times10^{4}
\end{aligned}\]

** You will now have two numbers with the same exponent**

6\times10^4 and 0.2\times10^4

You now add the two non-zero numbers

\[6 + 0.2 = 6.2\]

**The exponents are the same after what you did in step 1**

So your answer in standard form is 6.2\times10^4 .

Calculate (5\times10^{4})−(4\times10^{3}) .

Write your answer in standard form.

**Convert the number with the smaller power so that both numbers have the same power**

To do this identify the number with the lowest exponent.

\[4\times10^{3}\]

You need to increase the power. To do this multiply it by 10 to add one to the power. But to maintain the value of the number you need to divide the non-zero number by 10.

\[\begin{aligned}
&\quad \quad \quad\quad \;\; 4\times10^{3} \\\\
&\div10 ↓ \quad \quad \quad \quad \quad \quad \times10 ↓\\\\
&\quad \quad \quad \quad \;\; 0.4\times10^{4}
\end{aligned}\]

** You will now have two numbers with the same exponent**

5\times10^{4} and 0.4\times10^{4}

You now add the two non-zero numbers

\[5 – 0.4 = 4.6\]

**The exponents are the same after what you did in step 1**

So your answer in standard form is 4.6\times10^4 .

**Writing a number with the incorrect power for a large or small number**

This can happen by counting the zeros following the first non zero digit for large numbers or zeros after the decimal point for small numbers, then writing this as the power.

**Identifying incorrect place value with small numbers**

For example, in a number such as

**Not converting solutions to standard form**

After calculating with standard form, a common mistake is to not check that the first part of the number is

**Not giving the solution in the correct form**

It is important to check what form the question asks for the solution in – ordinary number or standard form.

1. Write 86,000 in standard form.

8.6 \times 10^{3}

8.6 \times 10^{4}

86 \times 10^{3}

8.6 \times 10^{-4}

The number between 1 and 10 here is 8.6. Since 8 is in the ten thousands column,

\begin{aligned} 86000&=8.6 \times 10000\\\\ &= 8.6 \times 10^{4} \end{aligned}

2. Write 0.0097 in standard form.

9.7 \times 10^{3}

9.7 \times 10^{-3}

9.7 \times 10^{-4}

0.97 \times 10^{-2}

The number between 1 and 10 here is 9.7. Since 9 is in the thousandths position,

\begin{aligned} 0.0097 &= 9.7 \times \frac{1}{1000}\\\\ &= 9.7 \times 10^{-3} \end {aligned}

3. Write 5.9 \times 10^{3} as an ordinary number.

59000

590

5900

0.0059

10^{3}=1000

Therefore

\begin{aligned} 5.9 \times 10^{3} &= 5.9 \times 1000\\\\ &=5900 \end{aligned}

4. Work out (6\times10^{8})\times(3\times10^{4}) . Write your answer in standard form.

18 \times 10^{12}

18 \times 10^{32}

1.8 \times 10^{11}

1.8 \times 10^{13}

\begin{aligned}
&6 \times 3=18\\\\
&10^{8} \times 10^{4} = 10^{12}
\end{aligned}

Therefore

(6 \times 10^{8}) \times (3 \times 10^{4}) = 18 \times 10^{12}

However 18 \times 10^{12} is not in standard form since 18 is not between 1 and 10.

So we need to divide 18 by 10 and, to compensate, increase the power of ten by 1.

This gives us

1.8 \times 10^{13}

5. Work out (9\times10^{7}) \div (4\times10^{2}) . Write your answer in standard form.

2.25 \times 10^{5}

22.5 \times 10^{4}

0.225 \times 10^{6}

2.25 \times 10^{3.5}

\begin{aligned}
&9 \div 4 = 2.25\\\\
&10^{7} \div 10^{2}=10^{5}
\end{aligned}

Therefore

(9 \times 10^{7}) \div (4 \times 10^{2}) = 2.25 \times 10^{5}

This is already in standard form since

1 \leq 2.25 <10

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

7.2 \times 10^{5}

7.2 \times 10^{4}

72 \times 10^{4}

9 \times 10^{9}

2 \times 10^{4} = 0.2 \times 10^{5}

(increase the power of ten by 1 and divide the 2 by 10 to compensate)

(7.2 \times 10^{5}) + ( 0.2 \times 10^{5}) = 7.2 \times 10^{5}

1. The distance from Earth to Mars is 2.88\times10^8km, to three significant figures. Write this number in standard form.

**(1 Mark)**

Show answer

288, 000, 000 km

**(1)**

2. (a) The speed of light is 3\times10^8m/s , rounded to one significant figure. Write this as an ordinary number.

(b) The mass of Earth is 5.97\times10^{24}kg . Work out the mass of the Earth in grams. Write your answer in standard form.

**(3 Marks)**

Show answer

(a)

300, 000, 000 m/s

**(1)**

(b)

Recognising 1000g in a kilogram or multiplying by 1\times10^3

**(1)**

5.97\times10^{27} g

**(1)**

3. Put these numbers in order. Start with the smallest number.

62\times10^{-2}, \quad \quad 0.0068, \quad \quad 6.9\times10^{-3}, \quad \quad 0.0607

**(2 Marks)**

Show answer

Converting all of the numbers to the same form for comparison or orders 3 of the four numbers correctly.

**(1)**

0.0068, \quad \quad 6.9\times10^{-3}, \quad \quad 0.0607, \quad \quad 62\times10^{-2}

**(1)**

You have now learned how to:

- Write numbers in standard form
- Calculate with numbers in standard form

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