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

Answer 1

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

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