Which expression is equivalent to \( \left(5^{-1}\right)^{2} ? \) Answer \( \frac{1}{25} \) 5 \( \frac{1}{125} \)

To solve \( \left(5^{-1}\right)^{2} \), we can apply the power of a power property which states that \( (a^m)^n = a^{m \cdot n} \). Here, \( a = 5 \), \( m = -1 \), and \( n = 2 \). Thus, \( \left(5^{-1}\right)^{2} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \). Now, speaking of exponents, did you know that negative exponents represent the reciprocal? This translates to \( a^{-n} = \frac{1}{a^n} \). So each time you see a negative exponent, think of turning it into a fraction! Lastly, ensuring we correctly apply exponent rules is critical. A common mistake is computing powers incorrectly, especially when negative signs are involved. Always double-check your exponent calculations to avoid this pitfall!