1.4.1 Try some yourself
Activity 8
If a litre is one cubic decimetre, how many litres are there in a cubic metre?
Answer
Since a litre is one cubic decimeter, the question is asking how may cubic decimetres are in a cubic metre.
1 metre = 10 decimetres
so 1 m^{3} = 10^{3} cubic decimetres.
Hence there are 1000 litres in a cubic metre.
Activity 9
Find the following without using your calculator, as an estimate for the calculator work in the next question.
(a) (^{−}1)^{3}
(b) 3^{3}
(c) 100^{3}
(d) 0.1^{3}
Answer
(a) (^{−}1)^{3} = ^{−}1 × ^{−}1 × ^{−}1 = 1 × ^{−}1 = ^{−}1
(b) 3^{3} = 3 × 3 × 3 = 9 × 3 = 27
(c) 100^{3} = 100 × 100 × 100 = 10 000 × 100 = 1000 000 (a million)
(d)
Activity 10
Use your calculator to find the following.
(a) (^{−}1.2)^{3}
(b) 3.3^{3}
(c) 101^{3}
(d) 0.121^{3}
Answer

(a) Estimate: (^{−}1)^{3} = ^{−}1.
Calculate: (^{−}1.2)^{3} = ^{−}1.728.

(b) Estimate: 3^{3} = 27.
Calculate: 3.3^{3} = 35.937.

(c) Estimate: 100^{3} = 1000 000.
Calculate: 101^{3} = 1030 301.

(d) Estimate: .1^{3} = .001.
Calculate: 0.121^{3} = 0.00177 (3 s.f.)
Activity 11
What are the following?
(a)
(b)
Answer
(a) = 1 since 1^{3} = 1.
(b) = 10 since 10^{3} = 1000.
Activity 12
Find the volume of a onefoot cube in cubic metres (1 foot = 30.48 cm). Estimate your answer first. Round your answer to three decimal places.
Answer
To estimate an answer, choose a simple approximation, say
1 foot 30 cm = 0.3 m.
Then 1 foot cubed (0.3)^{3} = 0.027 m^{3}.
More accurately, 1 foot = 30.48 cm = 0.3048 m.
So 1 foot cubed = (0.3048)^{3} = 0.028 m^{3} to three d.p.
Activity 13
Without using your calculator, find the following, as estimates for the calculator work in Question 7.
(a) 9^{2}
(b) 4^{3}
(c)
(d)
(e) (^{−}3)^{2}
Answer
(a) 9^{2} = 9 × 9 = 81
(b) 4^{3} = 4 × 4 × 4 = 16 × 4 = 64
(c) = 8 (since 8^{2} = 64)
(d)
(e) (^{−}3)^{2} = ^{−}3 × ^{−}3 = 9
Activity 14
Use your calculator to find:
(a) 9.42^{2}
(b) 3.65^{3}
(c)
(d) 0.33^{3}
(e) (^{−}2.713)^{2}
Use your answers to the previous question as rough checks. Round your answers to four decimal places.
Answer
(a) 9.42^{2} = 88.7364
(b) 3.65^{3} 48.6271
(c) 8.3666
(d) 0.33^{3} 0.0359 (check this by calculating , the estimate from 3(d) above, as a decimal.
(e) (^{−}2.713)^{2} 7.3604 (N.B. To get this answer on a calculator, remember the brackets.)