+57
Let x be a real number such that 625^x = 64.
Then 125^x = \(?\sqrt?\)
We can use the fact that $625=5^4$ and $64=2^6$ to rewrite the given equation as:
(5^4)^x = 2^6
Simplifying the left-hand side using the power of a power rule gives:
5^(4x) = 2^6
We can solve for $x$ by taking the logarithm of both sides with respect to 4, 2 or any other base:
\begin{align*} 4x\log_5(5) &= 6\log_5(2) \ 4x &= \frac{6}{\log_5(5) - \log_5(2)} \ 4x &= \frac{6}{1 - \log_2(5)} \ x &= \frac{3}{2(1-\log_2(5))} \end{align*}
Now, to find $125^x$, we can use the fact that $125=5^3$ and substitute the value we found for $x$:
125^x = (5^3)^(3/(2(1 - log_2(5))) = 5^(9/2(1 - log_2(5)))
Therefore, the value of 125^x is:
5^(9/2(1 - log_2(5))) = 10*sqrt(5)
