Which of the following is an equivalent expression to 5 ^9.5^-13 with only positive exponents that has been generated by applying the Properties of Integer Exponents?

Answer 1

To rewrite the expression \( 5^{9.5^{-13}} \) using only positive exponents, we need to analyze the exponent \( 9.5^{-13} \).

1. Recognize that \( 9.5^{-13} = \frac{1}{9.5^{13}} \). We can apply the property of exponents which states that \( a^{-b} = \frac{1}{a^b} \).

2. So, we have: \[ 5^{9.5^{-13}} = 5^{\frac{1}{9.5^{13}}} \]

3. To keep the exponent positive, we can rewrite this without negative exponents as: \[ 5^{(9.5^{-13})} = 5^{\frac{1}{9.5^{13}}} \]

   While this doesn't provide a straightforward simplification, we can recognize 9.5 as a decimal. Importantly, \( 9.5 \) can be expressed as \( \frac{19}{2} \). Applying this, the exponent becomes \( (9.5)^{-13} = \left( \frac{19}{2} \right)^{-13} = \frac{2^{13}}{19^{13}}\).

4. Substituting this back into the expression results in: \[ 5^{(9.5^{-13})} = 5^{(2^{13}/19^{13})} \]

In conclusion, the expression \( 5^{(9.5^{-13})} \) represents the required equivalent expression written with positive exponents. The original form would be the simplest way of showing this without any negative exponents directly applied within \( 5^{9.5^{-13}} \).