Evaluate:
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) $125 \sqrt[3]{\alpha^{6}}-\sqrt[3]{125 \alpha^{6}}$

Solution:

Property:

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

(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:​

$\sqrt[3]{700 \times 2 \times 49 \times 5}$

$=\sqrt[3]{2 \times 2 \times 5 \times 5 \times 7 \times 2 \times 7 \times 7 \times 5} \quad(\because 700=2 \times 2 \times 5 \times 5 \times 7$ and $49=7 \times 7)$

$=\sqrt[3]{2^{3} \times 5^{3} \times 7^{3}}$

$=\sqrt[3]{2^{3}} \times \sqrt[3]{5^{3}} \times \sqrt[3]{7^{3}}$

$=2 \times 5 \times 7$

$=70$

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

$125 \sqrt[3]{a^{6}}-\sqrt[3]{125 a^{6}}$

$125 \sqrt[3]{a^{6}}-\sqrt[3]{125 a^{6}}$

$=125 \sqrt[3]{a^{6}}-\left(\sqrt[3]{125} \times \sqrt[6]{a^{6}}\right)$

$=125 \times a^{2}-\left(5 \times a^{2}\right) \quad\left(\because \sqrt[3]{a^{6}}=\sqrt[3]{[a \times a \times a\} \times\{a \times a \times a\}}=a \times a=a^{2}\right.$ and $\left.\sqrt[3]{125}=\sqrt[3]{5 \times 5 \times 5}=5\right)$

$=125 a^{2}-5 a^{2}$

$=120 a^{2}$