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Evaluate each of the following:
(i) $ \sqrt[3]{27}+\sqrt[3]{0.008}+\sqrt[3]{0.064} $
(ii) $ \sqrt[3]{1000}+\sqrt[3]{0.008}-\sqrt[3]{0.125} $
(iii) $ \sqrt[3]{\frac{729}{216}} \times \frac{6}{9} $
(iv) $ \sqrt[3]{\frac{0.027}{0.008}} \div \sqrt{\frac{0.09}{0.04}}-1 $
(v) $ \sqrt[3]{0.1 \times 0.1 \times 0.1 \times 13 \times 13 \times 13} $
To find:
We have to evaluate the given expressions.
Solution:
(i) $\sqrt[3]{27}+\sqrt[3]{0.008}+\sqrt[3]{0.064}= \sqrt[3]{3 \times 3 \times 3}+\sqrt[3]{\frac{8}{1000}}+\sqrt[3]{\frac{64}{1000}}$
$=\sqrt[3]{3^{3}}+\sqrt[3]{\frac{2 \times 2 \times 2}{10 \times 10 \times 10}}+\sqrt[3]{\frac{4 \times 4 \times 4}{10 \times 10 \times 10}}$
$=3+\frac{2}{10}+\frac{4}{10}$
$=3+0.2+0.4$
$=3.6$
(ii) $\sqrt[3]{1000}+\sqrt[3]{0.008}-\sqrt[3]{0.125}=\sqrt[3]{10 \times 10 \times 10}+\sqrt[3]{\frac{8}{1000}}-\sqrt[3]{\frac{125}{1000}}$
$=\sqrt[3]{10^{3}}+\sqrt[3]{\frac{2 \times 2 \times 2}{10 \times 10 \times 10}}-\sqrt[3]{\frac{5 \times 5 \times 5}{10 \times 10 \times 10}}$
$=\sqrt[3]{10^{3}}+\sqrt[3]{\frac{2^{3}}{10^{3}}}-\sqrt[3]{\frac{5^{3}}{10^{3}}}$
$=10+\frac{2}{10}-\frac{5}{10}$
$=10+0.2-0.5$
$=10.2-0.5$
$=9.7$
(iii) $\sqrt[3]{\frac{729}{216}} \times \frac{6}{9}=\frac{\sqrt[3]{729}}{\sqrt[3]{216}} \times \frac{6}{9}$
$=\frac{\sqrt[3]{9 \times 9 \times 9}}{\sqrt[3]{6 \times 6 \times 6}} \times \frac{6}{9}$
$=\frac{\sqrt[3]{9^{3}}}{\sqrt[3]{6^{3}}} \times \frac{6}{9}$
$=\frac{9}{6} \times \frac{6}{9}$
$=1$
(iv) $\sqrt[3]{\frac{0.027}{0.008}} \div \sqrt{\frac{0.09}{0.04}}-1=\frac{\sqrt[3]{0.027}}{\sqrt[3]{0.008}} \div \sqrt{\frac{0.09}{0.04}}-1$
$=\frac{\sqrt[3]{\frac{27}{1000}}}{\sqrt[3]{\frac{8}{1000}}} \div \frac{\sqrt{\frac{9}{100}}}{\sqrt{\frac{4}{100}}}-1$
$=\frac{\sqrt[3]{\frac{27}{1000}}}{\sqrt[3]{\frac{8}{1000}}} \div \frac{\sqrt{\frac{9}{100}}}{\sqrt{\frac{4}{100}}} -1$
$=\frac{\frac{\sqrt[3]{27}}{\sqrt[3]{1000}}}{\frac{\sqrt[3]{8}}{\sqrt[3]{1000}}} \div \frac{\frac{\sqrt{9}}{\sqrt{100}}}{\frac{\sqrt{4}}{\sqrt{100}}} -1$
$=\frac{\frac{\sqrt[3]{3^{3}}}{\sqrt[3]{10^{3}}}}{\frac{\sqrt[3]{8^{3}}}{\sqrt[3]{10^{3}}}} \div \frac{\frac{\sqrt{3^{2}}}{\sqrt{10^{2}}}}{\frac{\sqrt{2^{2}}}{\sqrt{100}}} -1$
$=\frac{\frac{3}{10}}{\frac{2}{10}} \div \frac{\frac{3}{10}}{\frac{2}{10}} -1$
$=\frac{3}{2} \times \frac{2}{3}-1$
$=1-1$
$=0$
(v) $\sqrt[3]{0.1 \times 0.1 \times 0.1 \times 13 \times 13 \times 13}$
$=\sqrt[3]{(0.1)^{3} \times(13)^{3}}$
$=0.1 \times 13$
$=1.3$
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