Fill in the Blanks:
(i) $ \sqrt[3]{125 \times 27}=3 \times $
(ii) $ \sqrt[3]{8 \times \ldots}=8 $
(iii) $ \sqrt[3]{1728}=4 \times $__
(iv) $ \sqrt[3]{480}=\sqrt[3]{3} \times 2 \times \sqrt[3]{-} $
(v) $ \sqrt[3]{\square}=\sqrt[3]{7} \times \sqrt[3]{8} $
(vi) $ \sqrt[3]{-}=\sqrt[3]{4} \times \sqrt[3]{5} \times \sqrt[3]{6} $
(vii) $ \sqrt[3]{\frac{27}{125}}=\frac{}{5} $
(viii) $ \sqrt[3]{\frac{729}{1331}}=\frac{9}{-} $
(ix) $ \sqrt[3]{\frac{512}{-}}=\frac{8}{13} $

To find: 

We have to fill in the blanks.

Solution:

(i)  $\sqrt[3]{125 \times 27}=\sqrt[3]{5 \times 5 \times 5 \times 3 \times 3 \times 3}$

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

$=5 \times 3$

$=3 \times 5$

(ii)  $\sqrt[3]{8 \times}$__$=8$

$\sqrt[3]{8 \times}$__$=\sqrt[3]{8^{3}}$

$=\sqrt[3]{8 \times 8 \times 8}$

Therefore,

$\sqrt[3]{8 \times 8 \times 8}=8$

(iii) $\sqrt[3]{1728}=\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3}$

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

$=2 \times 2 \times 3$

$=4 \times 3$

Therefore,

$\sqrt[3]{1728}=4 \times \underline{3}$

(iv) $\sqrt[3]{480}=\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5}$

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

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

$=2 \times \sqrt[3]{3} \times \sqrt[3]{20}$

Therefore,

$\sqrt[3]{480}=\sqrt[3]{3} \times 2 \times \underline{\sqrt[3]{20}}$

(v) $\sqrt[3]{\square}=\sqrt[3]{7} \times \sqrt[3]{8}$

$\sqrt[3]{7} \times \sqrt[3]{8}=\sqrt[3]{7 \times 8}$

$=\sqrt[3]{56}$

Therefore,

$\sqrt[3]{\underline{56}}=\sqrt[3]{7} \times \sqrt[3]{8}$

(vi) $\sqrt[3]{4} \times \sqrt[3]{5} \times \sqrt[3]{6}=\sqrt[3]{4 \times 5 \times 6}$

$=\sqrt[3]{120}$

Therefore,

$\sqrt[3]{\underline{120}}=\sqrt[3]{4} \times \sqrt[3]{5} \times \sqrt[3]{6}$ 

(vii) $\sqrt[3]{\frac{27}{125}}=\sqrt[3]{\frac{3 \times 3 \times 3}{5 \times 5 \times 5}}$

$=\sqrt[3]{\frac{3}{5} \times \frac{3}{5} \times \frac{3}{5}}$

$=\sqrt[3]{(\frac{3}{5})^{3}}$

$=\frac{\underline{3}}{5}$

(viii) $\sqrt[3]{\frac{729}{1331}}=\sqrt[3]{\frac{9 \times 9 \times 9}{11 \times 11 \times 11}}$

$=\sqrt[3]{\frac{9^{3}}{11^{3}}}$

$=\frac{9}{11}$

Therefore,

$\sqrt[3]{\frac{729}{1331}}=\frac{9}{11}$

(ix) $\frac{8}{13}=\sqrt{(\frac{8}{13})^{3}}$

$=\sqrt[3]{\frac{8 \times 8 \times 8}{13 \times 13 \times 13}}$

$=\sqrt[3]{\frac{512}{2197}}$

Therefore,

$\sqrt[3]{\frac{512}{2197}}=\frac{8}{13}$