Question

Simplify using the laws of exponents. 25 -1/2 5 1/125 1/5 -125

Answer 1

To simplify the expression \( 25^{-1/2} \cdot 5^{1/125} \cdot \frac{1}{5^{-125}} \), we will first break down each term:

1. **Simplifying \( 25^{-1/2} \)**:
- \( 25 = 5^2 \)
- Thus, \( 25^{-1/2} = (5^2)^{-1/2} = 5^{-1} = \frac{1}{5} \)

2. **Simplifying \( 5^{1/125} \)**:
- This term remains as is: \( 5^{1/125} \)

3. **Simplifying \( \frac{1}{5^{-125}} \)**:
- \( \frac{1}{5^{-125}} = 5^{125} \) (because \( \frac{1}{a^{-n}} = a^{n} \))

Now, substituting these simplified forms back into the expression:

\[
\frac{1}{5} \cdot 5^{1/125} \cdot 5^{125}
\]

Next, we will combine the powers of 5 using the product rule of exponents, which states that \( a^m \cdot a^n = a^{m+n} \):

\[
5^{-1} \cdot 5^{1/125} \cdot 5^{125} = 5^{-1 + \frac{1}{125} + 125}
\]

Next, we need to combine the exponents:

\[
-1 + \frac{1}{125} + 125 = -\frac{125}{125} + \frac{1}{125} + \frac{125 \cdot 125}{125} = \frac{-125 + 1 + 15625}{125} = \frac{15501}{125}
\]

Finally, our expression simplifies to:

\[
5^{\frac{15501}{125}}
\]

This is the final simplified version of the given expression.