Show that:
(i) $ \sqrt[3]{27} \times \sqrt[3]{64}=\sqrt[3]{27 \times 64} $
(ii) $ \sqrt[3]{64 \times 729}=\sqrt[3]{64} \times \sqrt[3]{729} $
(iii) $ \sqrt[3]{-125 \times 216}=\sqrt[3]{-125} \times \sqrt[3]{216} $
(iv) $ \sqrt[3]{-125 \times-1000}=\sqrt[3]{-125} \times \sqrt[3]{-1000} $

To find: 

We have to show that:

(i) \( \sqrt[3]{27} \times \sqrt[3]{64}=\sqrt[3]{27 \times 64} \)

(ii) \( \sqrt[3]{64 \times 729}=\sqrt[3]{64} \times \sqrt[3]{729} \)

(iii) \( \sqrt[3]{-125 \times 216}=\sqrt[3]{-125} \times \sqrt[3]{216} \)

(iv) \( \sqrt[3]{-125 \times-1000}=\sqrt[3]{-125} \times \sqrt[3]{-1000} \)

Solution:

(i) LHS $=\sqrt[3]{27} \times \sqrt[3]{64}$

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

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

$=3 \times 4$

$=12$

RHS $=\sqrt[3]{27 \times 64}$

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

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

$=3 \times 4$

$=12$

LHS $=$ RHS

Hence proved.

(ii) LHS $=\sqrt[3]{64 \times 725}$

$=\sqrt[3]{4 \times 4 \times 4 \times 9 \times 9 \times 9}$

$=\sqrt[3]{4^{3} \times 9^{3}}$

$=4 \times 9$

$=36$

RHS $=\sqrt[3]{64} \times \sqrt[3]{729}$

$=\sqrt[3]{4 \times 4 \times 4}\times\sqrt[3]{9 \times 9 \times 9}$

$=4\times9$

$=36$

LHS $=$ RHS

Hence proved.

(iii) LHS $=-\sqrt[3]{125 \times 216}$

$=-\sqrt[3]{5 \times 5 \times 5 \times 6 \times 6 \times 6}$

$=- \sqrt[3]{5^{3} \times 6^{3}}$

$=-5 \times 6$

$=-30$

RHS $=\sqrt[3]{-125} \times \sqrt[3]{216}$

$=-\sqrt[3]{5 \times 5 \times 5} \times \sqrt[3]{6 \times 6 \times 6}$

$=-\sqrt[3]{5^{3}} \times \sqrt[3]{6^{3}}$

$=-5 \times 6$

$=-30$

LHS $=$ RHS

Hence proved. 

(iv) LHS $=\sqrt[3]{-125 \times-1000}$

$=\sqrt[3]{125 \times 1000}$

$=\sqrt[3]{5 \times 5 \times 5 \times 10 \times 10 \times 10}$

$=\sqrt[3]{5^{3} \times 10^{3}}$

$=5 \times 10$

$=50$

RHS $=\sqrt[3]{-125} \times \sqrt[3]{-1000}$

$=(-\sqrt[3]{125}) \times(-\sqrt[3]{1000})$

$=(-\sqrt[3]{5 \times 5 \times 5}) \times(-\sqrt[3]{10 \times 10 \times 10})$

$=(-\sqrt[3]{5^{3}}) \times(-\sqrt[3]{10^{3}})$

$=(-5) \times(-10)$

$=50$

LHS $=$ RHS

Hence proved.