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.