Question

Evaluate:
(i) $\sqrt[3]{4^3\times 6^3}$
(ii) $\sqrt[3]{8\times 17\times 17\times 17}$
(iii) $\sqrt[3]{700\times 2\times 49\times 5}$
(iv) $125\sqrt[3]{{\alpha }^6}-\sqrt[3]{125{\alpha }^6}$

Answer 1

Property:

For any two integers a and b, $\sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}$

(i) From the above property, we have:

$\sqrt[3]{4^3\times 6^3}=\sqrt[3]{4^3}\times \sqrt[3]{6^3}=4\times 6=24$

(ii) Use above property and proceed as follows:

$\sqrt[3]{8\times 17\times 17\times 17}=\sqrt[3]{2^3\times {17}^3}=\sqrt[3]{2^3}\times \sqrt[3]{{17}^3}=2\times 17=34$

(iii) From the above property, we have:​

$\sqrt[3]{700\times 2\times 49\times 5}$
$=\sqrt[3]{2\times 2\times 5\times 5\times 7\times 2\times 7\times 7\times 5}$ ($\because$ $700=2\times 2\times 5\times 5\times 7\text{and}49=7\times 7$)
$$\begin{gathered} =\sqrt[3]{2^3\times 5^3\times 7^3} \\ =\sqrt[3]{2^3}\times \sqrt[3]{5^3}\times \sqrt[3]{7^3} \\ =2\times 5\times 7 \\ =70 \end{gathered}$$

(iv)

From the above property, we have:​

$$\begin{gathered} 125\sqrt[3]{a^6}-\sqrt[3]{125a^6} \\ =125\sqrt[3]{a^6}-\left(\sqrt[3]{125}\times \sqrt[6]{a^6}\right) \end{gathered}$$
$=125\times a^2-\left(5\times a^2\right)$ ($\because$ $\sqrt[3]{a^6}=\sqrt[3]{\left\{a\times a\times a\right\}\times \left\{a\times a\times a\right\}}=a\times a=a^{2}and\sqrt[3]{125}=\sqrt[3]{5\times 5\times 5}=5$)
$$\begin{gathered} =125a^2-5a^2 \\ =120a^2 \end{gathered}$$