# 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) $\sqrt[3]{0.1 \times 0.1 \times 0.1 \times 13 \times 13 \times 13}$

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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) $\sqrt[3]{0.1 \times 0.1 \times 0.1 \times 13 \times 13 \times 13}$

$=\sqrt[3]{(0.1)^{3} \times(13)^{3}}$

$=0.1 \times 13$

$=1.3$

Updated on 10-Oct-2022 12:47:25