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.

earth and sun

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 104 = 3 × 10 000 = 30 000 (Since 104 = 10 × 10 × 10 × 10 = 10 000)

2 850 000 = 2.85 × 1 000 000 = 2.85 × 106 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 × 104 + 6.45 × 105

= 45,000 + 645,000

= 690,000

= 6.9 × 105

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, 105 × 103 = 108
  • To divide powers you subtract, eg, 105 ÷ 103 = 102

Example 2

Simplify (2 × 103) × (3 × 106)

Solution

Multiply 2 by 3 and add the powers of 10:

(2 × 103) × (3 × 106) = 6 × 109

Question

Simplify (36 × 105) ÷ (6 × 103)

toggle answer

Answer

Did you get 6 ×102?

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 105) ÷ (6 × 103) = (36 ÷ 6) x (105 ÷ 103) = 6 ×102

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.

 

PowerAnswerExamples
a0140 = 11000 = 1370 = 1

Anything to the power 0 is equal to 1.

a-b

a1/2

a1/3

Activity

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

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