$\begin{array}{l}\frac{840}{0.000021}=\frac{8.4\times {10}^2}{2.1\times {10}^{-5}}\\ =\frac{8.4}{2.1}\times {10}^{2-\left(-5\right)}\\ =4\times {10}^7\end{array}$

Example 1:

Solution:


Example 2:

Solution:

Example 1:

Solution:



Example 2:
Solution:



Example 3:

Solution:
= 1.055 (4 s. f.)

Question 1:

Solution:

Question 2:
Express each of the following as a single numbers.
(a) 8.565 × 10^-5
(b) 1.304 × 10⁵
(c) 6.754 × 10^-6

Solution:
(d) 10352

Question 3:
Express each of the following in standard form.
(a) 376510
(b) 47865400
(c) 0.000507
(d) 0.00006408

Solution:

Question 4:

Solution:

Question 5:

Solution:

1. Standard form is a way of writing very large numbers or very small numbers in the form of A × 10ⁿ, where  $1\le A<10$  and n is a positive or a negative integer.

Example 2:
Express each of the following numbers in standard form.
(a) 63.4
(b) 2738
(c) 23000
(d) 428000000
(e) 0.0063
(f) 0.000000038

Solution:
(a) 63.4 = 6.34 × 10

(b) 2738 = 2.738 × 1000 = 2.738 × 10³

(c) 23000 = 2.3 × 10000 = 2.3 × 10⁴

(d) 428000000 = 4.28 × 100000000 = 4.28 × 10⁸

(e) 0.0063
$=6.3\times \frac{1}{1000}$
= 6.3 × 10^-3
= 3.8 × 10^-8

Example 1:
Round off the following numbers correct to three significant figures.
(a) 246 = 246 (3 s. f.)
(b) 2463 = 2460 (3 s. f.)
(c) 24632 = 24600 (3 s. f.)
(d) 0.00745 = 0.00745 (3 s. f.)
(e) 0.007453 = 0.00745 (3 s. f.)
(f) 0.007455 = 0.00746 (3 s. f.)
(g) 0.007403 = 0.00740 (3 s. f.)

Example 2:

Solution:
(a) 3500 (2 s. f.)
(b) 0.509 (3 s. f.)
(c) 30 (1 s. f.)
(d) 0.401 (3 s. f.)
(e) 0.68 (2 s. f.)
(f) 39.0 (3 s. f.)

Question 9:
$3.17\times {10}^{-8}-1.20\times {10}^{-9}=$

Solution:
$\begin{array}{l}3.17\times {10}^{-8}-1.20\times {10}^{-9}\\ =3.17\times {10}^{-8}-\left(0.120\times {10}^1\times {10}^{-9}\right)\\ =3.17\times {10}^{-8}-0.120\times {10}^{-8}\\ =\left(3.17-0.120\right)\times {10}^{-8}\\ =3.05\times {10}^{-8}\end{array}$

Question 10:
$\text{Express }\frac{0.096}{{\left(2\times {10}^3\right)}^3}\text{ in standard form}\text{.}$

Solution:
$\frac{0.096}{{\left(2\times {10}^3\right)}^3}=\frac{0.096}{8\times {10}^9}=1.2\times {10}^{-11}$

Question 11:
$\text{Express }0.000064-3.5\times {10}^{-6}\text{ in standard form}\text{.}$

Solution:
$\begin{array}{l}0.000064-3.5\times {10}^{-6}\\ =6.4\times {10}^{-5}-3.5\times {10}^{-6}\\ =6.4\times {10}^{-5}-0.35\times {10}^{-5}\\ =\left(6.4-0.35\right)\times {10}^{-5}\\ =6.05\times {10}^{-5}\end{array}$

Question 12:
$\begin{array}{l}200.7\times {10}^{11}\text{ is written as }2.007\times {10}^b\text{ in the standard form}\text{.}\\ \text{State the value of }b.\end{array}$

Solution:
$\begin{array}{l}200.7\times {10}^{11}=2.007\times {10}^2\times {10}^{11}\\ =2.007\times {10}^{13}\\ b=13\end{array}$

Question 13:
Find the product of 0.1985 and 0.5.
Round off the answer correct to two significant figures.

Solution:
0.1985 × 0.5
= 0.09925
= 0.10 (two significant figures)

Question 14:
The area of a rectangular public parking area is 7.2 km² . Its width is 2400 m. The length, in m, of the parking area is

Solution:
$\begin{array}{l}\text{Area = Length }\times \text{ Width}\\ \text{Length }\times \text{ 2}\text{.400 }km\text{ = 7}\text{.2 }k{m}^2\\ \text{Length}=\frac{7.2}{2.4}\\ =3km\\ =3000m\\ =3\times {10}^{3}m\end{array}$

Question 15:
It is given that 30 solid metal cylinders each with a radius of 70 cm and a height of 300 cm, are melted to make 40 identical solid spheres.
Find the volume, in cm³ , of each solid sphere.

Solution:
$\begin{array}{l}\text{Volume of 40 spheres}\\ \text{= Volume of 30 cylinders}\\ =30\times \frac{22}{7}\times {70}^2\times 300\\ =138\text{ 6}00000{\text{ cm}}^3\\ \\ \text{Volume of 1 sphere }=\frac{138\text{ 6}00000}{40}\\ =3\text{ 465 }000{\text{ cm}}^3\\ =3.465\times {10}^6{\text{ cm}}^3\\ =3.47\times {10}^6{\text{ cm}}^3\end{array}$