Write the correct answer in the following: The product [3] 2 . [4] 2 . [12] 32 equals.

Write the correct answer in the following: The product [3] 2 . [4…

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MCQ

Write the correct answer in the following: The product $\sqrt[3]{2}.\sqrt[4]{2}.\sqrt[12]{32}$ equals.

A
$\sqrt{2}$
✓
$2$
C
$\sqrt[12]{2}$
D
$\sqrt[12]{32}$

✓

Answer

Correct option: B.

$2$

$LCM$ of $3, 4$ and $12 = 12$
$\sqrt[3]{2}=\sqrt[12]{2^4}\ [\because\sqrt[\text{m}]{\text{a}}=\sqrt[\text{mn}]{\text{a}^\text{n}}]$
$\sqrt[4]{2}=\sqrt[12]{2^3}$
and $\sqrt[12]{32}=\sqrt[12]{2^5}$
$\therefore\text{product of }\sqrt[3]{2}.\sqrt[4]{2}.\sqrt[12]{2^3}.\sqrt[12]{2^5}=\sqrt[12]{2^4.2^3.2^5}$
$=12\sqrt{2^{4+3+5}}=\sqrt[12]{2^{12}}=2^{12\times\frac{1}{12}}=2\ [\because\sqrt[\text{m}]{\text{a}^\text{n}}=\text{a}^{\frac{\text{n}}{\text{m}}}]$

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