Question

Find the value of:

$\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}}}}$

$\sf{a).\;3^{31/64}}$
$\sf{b).\;3^{31/32}}$
$\sf{c).\;3^{1/64}}$
$\sf{d).\;None\;of\;these}$

Quality Answer only!!!

Answer 1

Answer:

$\large \dag$ Question :-

Find the value of

$\sqrt{3 \sqrt{3 \sqrt{3 \sqrt{3 \sqrt{3 \sqrt{3} } } } } }$

$\large \dag$ Answer :-

• Option D)None of these .

$\large \dag$ Solutions :-

$\sqrt{} (3( \sqrt{3 \sqrt{3 \sqrt{3 \sqrt{3.3 {}^{ \frac{1}{2} } } } } } )$

$= (3(3(3(3(3.3 { \frac{1}{2} }^{} ) {}^{ \frac{1}{2} }) {}^{ \frac{1}{2} } ) {}^{ \frac{1}{2} } ) {}^{ \frac{1}{2} } ) {}^{ \frac{1}{2} } )$

• Which can also be written as,

${3}^{1. \frac{1}{2} . \frac{1}{2} . \frac{1}{2}. \frac{1}{2} . \frac{1}{2} . \frac{1}{2}. } = {3}^{ \frac{63}{64} }$

$\large \dag$ Hope it helps u mate .

$\large \dag$ Thank you .

Answer 2

Need to evaluate:-

• $\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}}}}$

Solution:-

Inorder to solve this problem, we must know the exponential properties.

$\boxed{\begin{array}{l}\underline{\text{Law of Exponents :}}\\\\\bullet\:\:\sf\dfrac{a^m}{a^n} = a^{m - n}\\\\\bullet\:\:\sf{(a^m)^n = a^{mn}}\\\\\bullet\:\:\sf(a^m)(a^n) = a^{m + n}\\\\\bullet\:\:\sf\dfrac{1}{a^n} = a^{-n}\\\bullet \:\:\sf\sqrt[\sf n]{\sf a} = (a)^{\dfrac{1}{n}}\end{array}}$

We will solve this question by using these properties.

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3. {3}^{ \frac{1}{2} } }}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{ {3}^{ \frac{1}{2} + 1 } }}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{ {3}^{ \frac{3}{2} } }}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{3. {3}^{ \frac{3}{2}. \frac{1}{2}}}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{3. {3}^{ \frac{3}{4}}}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{ {3}^{ \frac{3}{4} + 1}}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3\sqrt{ {3}^{ \frac{7}{4} }}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3. {3}^{ \frac{7}{4}. \frac{1}{2}}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{3. {3}^{ \frac{7}{8}}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{ {3}^{ \frac{7}{8} + 1}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3\sqrt{ {3}^{ \frac{15}{8}}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3. {3}^{ \frac{15}{8}. \frac{1}{2}}}}}$

$: \implies\sf{\sqrt{3\sqrt{3. {3}^{ \frac{15}{16}}}}}$

$: \implies\sf{\sqrt{3\sqrt{{3}^{ \frac{31}{16}}}}}$

$: \implies\sf{\sqrt{3.{3}^{ \frac{31}{16}. \frac{1}{2}}}}$

$: \implies\sf{\sqrt{3.{3}^{ \frac{31}{32}}}}$

$: \implies\sf{\sqrt{{3}^{ \frac{31}{32} + 1}}}$

$: \implies\sf{\sqrt{{3}^{ \frac{63}{32} }}}$

$: \implies\sf{{3}^{ \frac{63}{32}. \frac{1}{2}}}$

$: \implies\sf{{3}^{ \frac{63}{64}}}$

Therefore,

$\boxed{\sf\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}}} = {3}^{ \frac{63}{64} } }$

Option (D) non of these is the correct answer.