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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.
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