Sure, let's simplify the given expression step-by-step.
The expression given is:
[tex]\[
\left(7 a^{-5}\right)^2
\][/tex]
1. Distribute the power of 2 to both the coefficient and the variable:
[tex]\[
\left(7\right)^2 \cdot \left(a^{-5}\right)^2
\][/tex]
2. Calculate the coefficient term:
[tex]\[
(7)^2 = 49
\][/tex]
3. Simplify the exponent on the variable:
When you raise a power to another power, you multiply the exponents. So for [tex]\(a^{-5}\)[/tex] raised to the power of 2,
[tex]\[
\left(a^{-5}\right)^2 = a^{-5 \times 2} = a^{-10}
\][/tex]
4. Combine these results:
[tex]\[
\left(7 a^{-5}\right)^2 = 49 \cdot a^{-10}
\][/tex]
5. Rewrite using positive exponents:
A negative exponent [tex]\(a^{-10}\)[/tex] can be rewritten as the reciprocal with a positive exponent:
[tex]\[
a^{-10} = \frac{1}{a^{10}}
\][/tex]
Therefore, the final simplified expression using positive exponents is:
[tex]\[
49 \cdot \frac{1}{a^{10}} = \frac{49}{a^{10}}
\][/tex]
So the simplified expression is:
[tex]\[
\left(7 a^{-5}\right)^2 = \frac{49}{a^{10}}
\][/tex]
