Maths

Powers and roots - Higher

In this Revision Bite we are going to look at standard index form and zero, negative and fractional powers.

# Standard index form

Standard index form is also known as standard form. It is very useful when writing very big or very small numbers.

In standard form, a number is always written as:
A × 10 ^{n}

A is always between 1 and 10.
n tells us how many places to move the decimal point.

**Example : **Write 15 000 000 in standard index form.

**Solution**

15 000 000 = 1.5 × 10 000 000

This can be rewritten as:

1.5 × 10 × 10 × 10 × 10 × 10 × 10 × 10

= 1. 5 × 10 ^{7}

You can convert from standard form to ordinary numbers, and back again. Have a look at these examples:

3 x 10^{4} = 3 × 10 000 = 30 000
(Since 10^{4} = 10 × 10 × 10 × 10 = 10 000)

2 850 000 = 2.85 × 1 000 000 = 2.85 × 10^{6}
Make the first number between 1 and 10.

0.000467 = 4.67 × 0.0001 = 4.67 × 10 ^{-4}

# Standard index form

## Adding and subtracting numbers in standard index form:

Convert them into ordinary numbers, do the calculation, then change them back if you want the answer in standard form.

**Example 1**

4.5 × 10^{4} + 6.45 × 10^{5}

= 45,000 + 645,000

= 690,000

= 6.9 × 10^{5}

## Multiplying and dividing numbers in standard form:

Here you can use the rules for multiplying and dividing powers. Remember these rules:

- To multiply powers you add, eg, 10
^{5} × 10^{3} = 10^{8} - To divide powers you subtract, eg, 10
^{5} ÷ 10^{3} = 10^{2}

**Example 2**

Simplify (2 × 10^{3}) × (3 × 10^{6})

**Solution**

Multiply 2 by 3 and add the powers of 10:

(2 × 10^{3}) × (3 × 10^{6}) = 6 × 10^{9}

- Question
Simplify (36 × 10^{5}) ÷ (6 × 10^{3})

- Answer
Did you get 6 ×10^{2}?

If not, remember that you should first work out 36 ÷ 6, then work subtract the powers of 10 (because it is division), like this:

(36 x 10^{5}) ÷ (6 × 10^{3}) = (36 ÷ 6) x (10^{5} ÷ 10^{3}) = 6 ×10^{2}

# Zero, negative and fractional powers

In the previous pages, we only looked at positive whole number powers. We can also find zero, negative and fractional powers. The rules below apply to these powers.

##

Power | Answer | Examples |
---|

a^{0} | 1 | 4^{0} = 1100^{0} = 137^{0} = 1Anything to the power 0 is equal to 1. |

a^{-b} | | |

a^{1/2} | | |

a^{1/3} | | |

## Activity

Have a quick game of 'Powers snap' to help get your head around all these rules.

**Now try a **Test Bite