Find the cube of:

Question:

Find the cube of:

(i) $\frac{7}{9}$

(ii) $-\frac{8}{11}$

(iii) $\frac{12}{7}$

(iv) $-\frac{13}{8}$

(v) $2 \frac{2}{5}$

(vi) $3 \frac{1}{4}$

(vii) $0.3$

(viii) $1.5$

(ix) $0.08$

(x) $2.1$

Solution:

(i)

$\because\left(\frac{m}{n}\right)^{3}=\frac{m^{3}}{n^{3}}$

$\therefore\left(\frac{7}{9}\right)^{3}=\frac{7^{3}}{9^{3}}=\frac{7 \times 7 \times 7}{9 \times 9 \times 9}=\frac{343}{729}$

(ii)

$\because\left(-\frac{m}{n}\right)^{3}=-\frac{m^{3}}{n^{3}}$

$\therefore\left(-\frac{8}{11}\right)^{3}=-\left(\frac{8}{11}\right)^{3}=-\left(\frac{8^{3}}{11^{3}}\right)=-\left(\frac{8 \times 8 \times 8}{11 \times 11 \times 11}\right)=-\frac{512}{1331}$

(iii)

$\because\left(\frac{m}{n}\right)^{3}=\frac{m^{3}}{n^{3}}$

$\therefore\left(\frac{12}{7}\right)^{3}=\frac{12^{3}}{7^{3}}=\frac{12 \times 12 \times 12}{7 \times 7 \times 7}=\frac{1728}{343}$

(iv)

$\because\left(-\frac{m}{n}\right)^{3}=-\frac{m^{3}}{n^{3}}$

$\therefore\left(-\frac{13}{8}\right)^{3}=-\left(\frac{13}{8}\right)^{3}=-\left(\frac{13^{3}}{8^{3}}\right)=-\left(\frac{13 \times 13 \times 13}{8 \times 8 \times 8}\right)=-\frac{2197}{512}$

(v)
We have:

$2 \frac{2}{5}=\frac{12}{5}$

Also, $\left(\frac{m}{n}\right)^{3}=\frac{m^{3}}{n^{3}}$

$\therefore\left(\frac{12}{5}\right)^{3}=\frac{12^{3}}{5^{3}}=\frac{12 \times 12 \times 12}{5 \times 5 \times 5}=\frac{1728}{125}$

(vi)
We have:

$3 \frac{1}{4}=\frac{13}{4}$

Also, $\left(\frac{m}{n}\right)^{3}=\frac{m^{3}}{n^{3}}$

$\therefore\left(\frac{13}{4}\right)^{3}=\frac{13^{3}}{4^{3}}=\frac{13 \times 13 \times 13}{4 \times 4 \times 4}=\frac{2197}{64}$

(vii)
We have:

$0.3=\frac{3}{10}$

Also, $\left(\frac{m}{n}\right)^{3}=\frac{m^{3}}{n^{3}}$

$\therefore\left(\frac{3}{10}\right)^{3}=\frac{3^{3}}{10^{3}}=\frac{3 \times 3 \times 3}{10 \times 10 \times 10}=\frac{27}{1000}=0.027$

(viii)
We have:

$1.5=\frac{15}{10}$

Also, $\left(\frac{m}{n}\right)^{3}=\frac{m^{3}}{n^{3}}$

$\therefore\left(\frac{15}{10}\right)^{3}=\frac{15^{3}}{10^{3}}=\frac{15 \times 15 \times 15}{10 \times 10 \times 10}=\frac{3375}{1000}=3.375$

(ix)
We have:

$0.08=\frac{8}{100}$

Also, $\left(\frac{m}{n}\right)^{3}=\frac{m^{3}}{n^{3}}$

$\therefore\left(\frac{8}{100}\right)^{3}=\frac{8^{3}}{100^{3}}=\frac{8 \times 8 \times 8}{100 \times 100 \times 100}=\frac{512}{1000000}=0.000512$

(x)
We have:

$2.1=\frac{21}{10}$

Also, $\left(\frac{m}{n}\right)^{3}=\frac{m^{3}}{n^{3}}$

$\therefore\left(\frac{21}{10}\right)^{3}=\frac{21^{3}}{10^{3}}=\frac{21 \times 21 \times 21}{10 \times 10 \times 10}=\frac{9261}{1000}=9.261$