Question

solve the question number 31

Answer 1

ANSWER WITH FULL EXPLANATION

First, find the value of $n$ in the first equation given. 
$\sqrt{5^{n}}=125$
⇒ $(5^{n})^{\frac{1}{2}}=125$    [∵ $\sqrt{x} = x^{\frac{1}{2}}$]
⇒ $5^{(n*\frac{1}{2})}=125$    [∵ $(x^{m})^{n}=x^{(m*n)}$]
⇒ $5^{\frac{n}{2}}=125$
⇒ $5^{\frac{n}{2}}=5^{3}$   [∵ $125=5^{3}$]
⇒ $\frac{n}{2}=3$    [If $x^{m}=x^{n}$, then $m=n$]
Multiplying both sides by 2. 
$\frac{n}{2}*2=3*2$
⇒ $n=6$

Now, we know that $n=6$. We have to substitute this value into the given expression $5^{\sqrt[n]{64}}$. 
$5^{\sqrt[n]{64}}$
$=5^{\sqrt[6]{64}}$    [Substituting $n=6$. ]
$=5^{2}$    [∵ $\sqrt[6]{64}=2$]
$=25$

∴ $5^{\sqrt[n]{64}}=25$

Hope this may help you. 

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