Question : If $A=\frac{\sqrt{0.0004} \times \sqrt[3]{0.000008}}{\sqrt[4]{16000} \times \sqrt[3]{125000} \times \sqrt[4]{810}}$ and $B=\frac{\sqrt[3]{0.729} \times \sqrt[4]{0.0016}}{\sqrt{0.16}}$, then what is $A \times B$?
Option 1: $6 \times 10^{–7}$
Option 2: $\frac{7}{4} \times 10^{–8}$
Option 3: $5 \times 10^{–8}$
Option 4: $\frac{7}{3} \times 10^{–7}$
Correct Answer: $6 \times 10^{–7}$
Solution :
Given: The values of $A=\frac{\sqrt{0.0004} \times \sqrt[3]{0.000008}}{\sqrt[4]{16000} \times \sqrt[3]{125000} \times \sqrt[4]{810}}$ and $B=\frac{\sqrt[3]{0.729} \times \sqrt[4]{0.0016}}{\sqrt{0.16}}$.
$A=\frac{\sqrt{0.0004} \times \sqrt[3]{0.000008}}{\sqrt[4]{16000} \times \sqrt[3]{125000} \times \sqrt[4]{810}}=\frac{\sqrt{(0.02)^2}\times\sqrt[3]{(0.02)^3}}{\sqrt[4]{2^4\times10^3}\times\sqrt[3]{(50)^3}\times{\sqrt[4]{3^4\times10}}}$
$=\frac{{0.02} \times {0.02}}{(2\times50\times3)\times{\sqrt[4]{(10)^3}}\times{\sqrt[4]{10}}}=\frac{{0.02} \times {0.02}}{300\times{{\sqrt[4]{(10)^3\times10}}}}$
⇒ $A=\frac{{0.02} \times {0.02}}{300\times10}=\frac{0.0004}{3000}$
$B=\frac{\sqrt[3]{0.729} \times \sqrt[4]{0.0016}}{\sqrt{0.16}}=\frac{\sqrt[3]{(0.9)^3} \times \sqrt[4]{(0.2)^4}}{\sqrt{(0.4)^2}}$
⇒ $B= \frac{0.9\times 0.2}{0.4}=\frac{0.9}{0.2}$
The value of $A \times B=\frac{0.0004}{3000}\times\frac{0.9}{0.2}$
$=\frac{4}{30000000}\times\frac{9}{2}=6\times 10^{–7}$
Hence, the correct answer is $6 \times 10^{–7}$.
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