Which of the following is an equivalent expression for 5^1/2/3^√125 when simplified using exponent properties?

Answer 1

To simplify the expression \( \frac{5^{1/2}}{3^{\sqrt{125}}} \), let's first simplify each component.

1. Simplify \( \sqrt{125} \):

   We know that \( 125 = 5^3 \), so: \[ \sqrt{125} = \sqrt{5^3} = 5^{3/2} \]

2. Rewrite \( 3^{\sqrt{125}} \):

   Using the result from step 1, we have: \[ 3^{\sqrt{125}} = 3^{5^{3/2}} \]

3. Now substitute back into the original expression:

   Our expression becomes: \[ \frac{5^{1/2}}{3^{5^{3/2}}} \]

4. Putting it all together:

   Thus, the expression \( \frac{5^{1/2}}{3^{\sqrt{125}}} \) simplifies to: \[ \frac{5^{1/2}}{3^{5^{3/2}}} \]

Keep in mind that there's no further simplification available here since we have an exponentiation in the denominator (like \( 3^{5^{3/2}} \)) that does not combine neatly with \( 5^{1/2} \).

Alternatively, you can keep it as: \[ 5^{1/2} \cdot 3^{-5^{3/2}} \]

Either form expresses the equivalent value of the original expression using properties of exponents.