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

To find: 

We have to 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) \( 125 \sqrt[3]{a^{6}}-\sqrt[3]{125 a^{6}} \)

Solution:

(i) $\sqrt[3]{4^{3} \times 6^{3}}=\sqrt[3]{4^{3}}\times\sqrt[3]{6^{3}}$

$=4 \times 6$

$=24$

(ii) $\sqrt[3]{8 \times 17 \times 17 \times 17}=\sqrt[3]{2 \times 2 \times 2 \times 17 \times 17 \times 17}$

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

$=2 \times 17$

$=34$

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

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

$=2 \times 5 \times 7$

$=70$

(iv) $125\sqrt[3]{a^{6}}-\sqrt[3]{125 a^{6}}=125 \sqrt[3]{a^{2} \times a^{2} \times a^{2}}-\sqrt[3]{5 \times 5 \times 5 \times a^{2} \times a^{2} \times a^{2}}$

$=125 \sqrt[3]{(a^{2})^{3}}-\sqrt[3]{5^{3} \times(a^{2})^{3}}$

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

$=120 a^{2}$