Solving for n in $\sqrt{5^n} = 625$
The problem requires finding the value of $n$ that satisfies the equation $\sqrt{5^n} = 625$. We can solve this by simplifying the equation using properties of exponents.
Step 1: Rewrite the square root
The square root of $5^n$ can be expressed using fractional exponents:
$ \sqrt{5^n} = (5^n)^{\frac{1}{2}} = 5^{\frac{n}{2}} $
Step 2: Express 625 as a power of 5
We need to find the power of 5 that equals 625:
$ 625 = 5 \times 5 \times 5 \times 5 = 5^4 $
Step 3: Equate the expressions
Now, substitute the simplified terms back into the original equation:
$ 5^{\frac{n}{2}} = 5^4 $
Step 4: Solve for n
Since the bases are the same (both are 5), the exponents must be equal:
$ \frac{n}{2} = 4 $
Multiply both sides by 2 to isolate $n$:
$ n = 4 \times 2 $
$ n = 8 $
Conclusion
The value of $n$ that satisfies the equation $\sqrt{5^n} = 625$ is 8.
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