Unit 3 Section 4 : Standard Form

Standard form is a convenient way of writing very large or very small numbers. It is used on a scientific calculator when a number is too large or too small to be displayed on the screen.

Before using standard form, we revise multiplying and dividing by powers of 10.

Example 1

Calculate:

(a) 3 × 104
(b) 3.27 × 103
(c) 3 ÷ 102
(d) 4.32 ÷ 104
These examples lead to the approach used for standard form, which is a reversal of the approach used in Example 1.
In standard form, numbers are written as
a × 10n
where 1 ≤ a < 10 and n is an integer.

Example 2

Write the following numbers in standard form:

(a) 5720
(b) 7.4
(c) 473 000
(d) 6 000 000
(e) 0.09
(f) 0.000621

Example 3

Calculate:

(a) (3 × 106) × (4 × 103)
(b) (6 × 107) ÷ (5 × 10–2)
(c) (3 × 104) + (2 × 105)

Note on Using Calculators

Your calculator will have a key EE or EXP for entering numbers in standard form.

For example, for 3.2 × 107, press

3 . 2 EXP 7

which will appear on your display like this:

3.2 07

Some calculators also display the ' × 10 ' part of the number, but not all do. You need to find out what your calculator displays. Remember, you must always write the ' × 10 ' part when you are asked to give an answer in standard form.

Exercises

Question 1
Calculate:
(a)6.21 × 1000
(b)8 × 103
(c)4.2 × 102
(d)3 ÷ 1000
(e)6 ÷ 102
(f)3.2 ÷ 103
(g)6 × 10–3
(h)9.2 × 10–1
(i)3.6 × 10–2
Question 2
Write each of the following numbers in standard form:
(a)200 ×
(b)8000 ×
(c)9 000 000 ×
(d)62 000 ×
(e)840 000 ×
(f)12 000 000 000 ×
(g)61 800 000 000 ×
(h)3 240 000 ×
Question 3
Convert each of the following numbers from standard form to the normal decimal notation:
(a)3 × 104
(b)3.6 × 104
(c)8.2 × 103
(d)3.1 × 102
(e)1.6 × 104
(f)1.72 × 105
(g)6.83 × 104
(h)1.25 × 106
(i)9.17 × 103
Question 4
Write each of the following numbers in standard form:
(a)0.0004 × 10
(b)0.008 × 10
(c)0.142 × 10
(d)0.0032 × 10
(e)0.00199 × 10
(f)0.000000062 × 10
(g)0.0000097 × 10
(h)0.00000000000021 × 10
Question 5
Convert the following numbers from standard form to the normal decimal format:
(a)6 × 10–2
(b)7 × 10–1
(c)1.8 × 10–3
(d)4 × 10–3
(e)6.2 × 10–3
(f)9.81 × 10–4
(g)6.67 × 10–1
(h)3.86 × 10–5
(i)9.27 × 10–7
Question 6
Without using a calculator, determine:
(a)(4 × 104) × (2 × 105) × 10
(b)(2 × 106) × (3 × 105) × 10
(c)(6 × 104) × (8 × 10–9) × 10
(d)(3 × 10–8) × (7 × 10–4) × 10
(e)(6.1 × 106) × (2 × 10–5) × 10
(f)(3.2 × 10–5) × (4 × 10–9) × 10
Question 7
Without using a calculator, determine:
(a)(9 × 107) ÷ (3 × 104) × 10
(b)(8 × 105) ÷ (2 × 10–2) × 10
(c)(6 × 10–2) ÷ (2 × 10–3) × 10
(d)(6 × 104) ÷ (3 × 10–6) × 10
(e)(4.8 × 1012) ÷ (1.2 × 103) × 10
(f)(3.6 × 108) ÷ (9 × 103) × 10
Question 8
Without a calculator, determine the following, giving your answers in both normal and standard form:
(a)(6 × 105) + (3 × 106) = × 10
(b)(6 × 102) + (9 × 103) = × 10
(c)6 × 105 – 1 × 104 = × 10
(d)8 × 10–2 + 9 × 10–3 = × 10
(e)6 × 10–4 + 8 × 10–3 = × 10
(f)6 × 10–4 – 3 × 10–5 = × 10
Question 9
Use a calculator to determine:
(a)(3.4 × 106) × (2.1 × 104) × 10
(b)(6 × 1021) × (8.2 × 10–11) × 10
(c)(3.6 × 105) × (4.5 × 107) × 10
(d)(8.2 × 1011) ÷ (4 × 10–8) × 10
(e)(1.92 × 106) × (3.2 × 10–11) × 10
(f)(6.2 × 1014)3 × 10
Question 10
The radius of the earth is 6.4 × 106 m. Giving your answers in standard form, correct to 3 significant figures, calculate the circumference of the earth in:
(a) × 10 m
(b) × 10 cm
(c) × 10 mm
(d) × 10 km
Question 11
Sir Isaac Newton (1642-1727) was a mathematician, physicist and astronomer.
In his work on the gravitational force between two bodies he found that he needed to multiply their masses together.
(a)
Work out the value of the mass of the Earth multiplied by the mass of the Moon.
Give your answer in standard form.
Mass of Earth = 5.98 × 1024 kg
Mass of Moon = 7.35 × 1022 kg
× 10
Newton also found that he needed to work out the square of the distance between the two bodies.
(b)
Work out the square of the distance between the Earth and the Moon.
Give your answer in standard form.
Distance between Earth and Moon = 3.89 × 105 km
× 10
Newton's formula to calculate the gravitational force (F) between two bodies is
F =
where G is the gravitational constant, m1 and m2 are the masses of the two bodies, and R is the distance between them.
(c)
Work out the gravitational force (F) between the Sun and the Earth using Newton's formula with information in the box below.
Give your answer in standard form.
m1 m2 = 1.19 × 1055 kg2
R2 = 2.25 × 1016 km2
G = 6.67 × 10–20
× 10
Question 12
(a)
Which of these statements is true?
(i) 4 × 103 is a larger number than 43 .
(ii) 4 × 103 is the same size as 43 .
(iii) 4 × 103 is a smaller number than 43 .
(b)
One of the numbers below has the same value as 3.6 × 104. Choose the number.
(i)363
(ii)364
(iii)(3.6 × 10)4
(iv)0.36 × 103
(v)0.36 × 105
(c)
One of the numbers below has the same value as 2.5 × 10–3. Choose the number.
(i)25 × 10–4
(ii)2.5 × 103
(iii)–2.5 × 103
(iv)0.00025
(v)2500
(d)
(2 × 102) × (2 × 102) can be written more simply as 4 × 104. Write the following values as simply as possible:
(i) (3 × 102) × (2 × 10–2) × 10
(ii) × 10