Find the cube of the following numbers

 \( 14^3 = 14 \times 14 \times 14 \)

\( = 2744\)

b) 9

\(9^3 = 9 \times 9 \times 9 \)

 \( = 729\) 

c) 2.3

\(2.3^3 = 2.3 \times 2.3 \times 2.3\)

\(=12.167\)

 Use Mathematical tables to find the reciprocal of 0.0247, hence evaluate \(\frac{\sqrt[3]{3.025}}{0.0247}\) Correct to 2 d.p (3 marks) 

\(\sqrt[3]{3.025} = 1.4462\)

\(\frac{1}{0.0247} \)

\(= (0.0247)^{-1}\)

\(= (2.47 \times 10^{-2})^{-1}\)

\(= 0.4049 \times 10^2\)

\(= 40.49\)

Use reciprocal, square and square root tables to evaluate, to 4 significant figures, the expression

\(
\sqrt{\frac{1}{24.56} + 4.346^2}
\)
\(
24.56 = 2.456 \times 10
\)
\(
4.346^2 = 18.89
\)
\(
\sqrt{\frac{1}{2.456 \times 10} + 18.89}
\)

\(
= \sqrt{\frac{1}{2.456} \times \frac{1}{10} + 18.89}
\)
\(
= \sqrt{\left( 0.4072 \times \frac{1}{10} \right) + 18.89}
\)
\(
= \sqrt{0.04072 + 18.89}
\)

\(
= \sqrt{18.93072}
\)
\(
= \sqrt{18.93}
\)
\(
= 4.351 \quad \text{(Answer)}
\)

Use squares, square roots and reciprocal tables to evaluate (3mks)
\(
3.045^2 + \frac{1}{\sqrt{49.24}}
\)

\(
3.045^2 = 9.272
\)

\(
\sqrt{49.24} = 7.017
\)

\(
\frac{1}{7.017} = 0.1425
\)


\(
9.272 + \frac{1}{7.017}
\)

\(
= 9.272 + 0.1425
\)

\(
= 9.4175
\)

Find the cubes of each of the following numbers:

(a) 15.3  (b) 2341  (c) 0.16  (d) 0.00472

Solutions

(a) \( (15.3)^3 \)  
\( = 15.3 \times 15.3 \times 15.3 \)  
\( = 1.53 \times 10 \times 1.53 \times 10 \times 1.53 \times 10 \)  
\( = (1.53)^3 \times (10)^3 \)  
\( = 3.582 \times 1\ 000 \)  
\( = 3\ 582 \)  

(b) \( (2\ 341)^3 \)  
\( = (2.341 \times 10^3)^3 \)  
\( = 2.341^3 \times 10^9 \)  
\( = 12.829 \times 10^9 \)  
\( = 1.2829 \times 10^1 \times 10^9 \)  
\( = 1.2829 \times 10^{10} \)  
\( \approx 1.283 \times 10^{10} \) (correct to 4 s.f.)  

(c) \( (0.16)^3 \)  
\( = (1.6 \times 10^{-1})^3 \)  
\( = 1.6^3 \times 10^{-3} \)  
\( = 4.096 \times 10^{-3} \)  
\( = 0.004096 \)  

(d) \( (0.00472)^3 \)  
\( = (4.72 \times 10^{-3})^3 \)  
\( = 4.72^3 \times 10^{-9} \)  
\( = 105.15 \times 10^{-9} \)  
\( = 1.0515 \times 10^2 \times 10^{-9} \)  
\( = 1.0515 \times 10^{-7} \)  
\( \approx 1.052 \times 10^{-7} \) (correct to 4 s.f.)