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