Question

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) $\sqrt[3]{\frac{0.027}{0.008}}÷\sqrt[]{\frac{0.09}{0.04}}-1$
(v) $\sqrt[3]{0.1\times 0.1\times 0.1\times 13\times 13\times 13}$

Answer 1

(i)
To evaluate the value of the given expression, we need to proceed as follows:

$\sqrt[3]{27}+\sqrt[3]{0.008}+\sqrt[3]{0.064}=\sqrt[3]{3\times 3\times 3}+\sqrt[3]{\frac{8}{1000}}+\sqrt[3]{\frac{64}{1000}}$
$=\sqrt[3]{3\times 3\times 3}+\frac{\sqrt[3]{8}}{\sqrt[3]{1000}}+\frac{\sqrt[3]{64}}{\sqrt[3]{1000}}$
$$\begin{gathered} =\sqrt[3]{3\times 3\times 3}+\frac{\sqrt[3]{2\times 2\times 2}}{\sqrt[3]{1000}}+\frac{\sqrt[3]{4\times 4\times 4}}{\sqrt[3]{1000}} \\ =3+\frac{2}{10}+\frac{4}{10} \\ =3+0.2+0.4 \\ =3.6 \end{gathered}$$

Thus, the answer is 3.6.

(ii)
To evaluate the value of the given expression, we need to proceed as follows:

$\sqrt[3]{1000}+\sqrt[3]{0.008}-\sqrt[3]{0.125}=\sqrt[3]{10\times 10\times 10}+\sqrt[3]{\frac{8}{1000}}-\sqrt[3]{\frac{125}{1000}}$
$=\sqrt[3]{10\times 10\times 10}+\frac{\sqrt[3]{8}}{\sqrt[3]{1000}}-\frac{\sqrt[3]{125}}{\sqrt[3]{1000}}$
$$\begin{gathered} =\sqrt[3]{10\times 10\times 10}+\frac{\sqrt[3]{2^3}}{\sqrt[3]{1000}}-\frac{\sqrt[3]{5^3}}{\sqrt[3]{1000}} \\ =10+\frac{2}{10}-\frac{5}{10} \\ =10+0.2-0.5 \\ =9.7 \end{gathered}$$

Thus, the answer is 9.7.

(iii)
To evaluate the value of the given expression, we need to proceed as follows:

$\sqrt[3]{\frac{729}{216}}\times \frac{6}{9}=\sqrt[3]{\frac{729}{216}}\times \frac{6}{9}=\frac{\sqrt[3]{729}}{\sqrt[3]{216}}\times \frac{6}{9}=\frac{\sqrt[3]{9\times 9\times 9}}{\sqrt[3]{2\times 2\times 2\times 3\times 3\times 3}}\times \frac{6}{9}=\frac{9}{2\times 3}\times \frac{6}{9}=\frac{{\cancel{9}}^1}{\cancel{6}}\times \frac{{\cancel{6}}^1}{\cancel{9}}=1$

Thus, the answer is 1.

(iv)
To evaluate the value of the expression, we need to proceed as follows:
$\sqrt[3]{\frac{0.027}{0.008}}÷\sqrt{\frac{0.09}{0.04}}-1=\sqrt[3]{\frac{{\displaystyle \frac{27}{1000}}}{{\displaystyle \frac{8}{1000}}}}÷\sqrt{\frac{{\displaystyle \frac{9}{100}}}{{\displaystyle \frac{4}{100}}}}-1=\sqrt[3]{\frac{27}{8}}÷\sqrt{\frac{9}{4}}-1=\frac{\sqrt[3]{27}}{\sqrt[3]{8}}÷\frac{\sqrt{9}}{\sqrt{4}}-1=\frac{3}{2}÷\frac{3}{2}-1=\frac{{\cancel{3}}^1}{\cancel{2}}\times \frac{{\cancel{2}}^1}{\cancel{3}}-1=1-1=0$
Thus, the answer is 0.

(v)
To evaluate the value of the expression, we need to proceed as follows:

$\sqrt[3]{0.1\times 0.1\times 0.1\times 13\times 13\times 13}=\sqrt[3]{\frac{1}{10}\times \frac{1}{10}\times \frac{1}{10}\times 13\times 13\times 13}=\sqrt[3]{\frac{13\times 13\times 13}{10\times 10\times 10}}=\frac{\sqrt[3]{13\times 13\times 13}}{\sqrt[3]{10\times 10\times 10}}=\frac{13}{10}=1.3$

Thus, the answer is 1.3.