Question

Find the value of this expression:-

$\sqrt[3]{3} \times \sqrt[4]{3} \times \sqrt[12]{243}$

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Answer 1

Answer :-

${\huge {\red {\pmb { 3}}}}$

Refer the attachment for full Answer (◕દ◕)

Used Concepts :-

• ${ \red { \bf { \sqrt[n]{x^{m}} ={\bigg( x \bigg)}^{\footnotesize \dfrac{m}{n}} }}}$

• ${ \red { \bf { a^{m} \times a^{n} \times a^{k}= a^{m+n+k}}}}$

Answer 2

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\sqrt[3]{3} \times \sqrt[4]{3} \times \sqrt[12]{243}$

$\rm \:  =  \: \sqrt[3]{3} \times \sqrt[4]{3} \times \sqrt[12]{3 \times 3 \times 3 \times 3 \times 3}$

$\rm \:  =  \: \sqrt[3]{3} \times \sqrt[4]{3} \times \sqrt[12]{ {3}^{5} }$

We know,

$\\ \boxed{\tt{ \sqrt[n]{x} = y \: \rm\implies \:x = {\bigg(y\bigg) }^{\dfrac{1}{n} }}}$

So, using this, we get

$\rm \:  =  \: {\bigg(3\bigg) }^{\dfrac{1}{3} } \times {\bigg(3\bigg) }^{\dfrac{1}{4} } \times {\bigg(3\bigg) }^{\dfrac{5}{12} }$

We know,

$\\ \boxed{\tt{ {x}^{m} \times {x}^{n} = {x}^{m + n}}}$

So, using this, we get

$\rm \:  =  \: {\bigg(3\bigg) }^{\dfrac{1}{3} + \dfrac{1}{4} + \dfrac{5}{12} }$

$\rm \:  =  \: {\bigg(3\bigg) }^{\dfrac{4 + 3 + 5}{12}}$

$\rm \:  =  \: {\bigg(3\bigg) }^{\dfrac{12}{12}}$

$\rm \:  =  \: {3}^{1}$

$\rm \:  =  \: 3$

Hence,

$\\ \rm\implies \:\boxed{\tt{ \sqrt[3]{3} \times \sqrt[4]{3} \times \sqrt[12]{243} = 3}}$

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MORE TO KNOW

$\boxed{\tt{ {x}^{0} = 1 \: }}$

$\boxed{\tt{ {x}^{m} \div {x}^{n} = {x}^{m - n}}}$

$\boxed{\tt{ {( {x}^{m}) }^{n} \: = \: {x}^{mn} }}$