To simplify the expression [tex]\(\left(6 a^{-5}\right)^3\)[/tex] and express the answer using positive exponents only, follow these steps:
1. Expand the expression inside the parentheses:
[tex]\[
(6 a^{-5})^3
\][/tex]
2. Use the property of exponents [tex]\((x \cdot y)^n = x^n \cdot y^n\)[/tex]:
[tex]\[
(6 a^{-5})^3 = 6^3 \cdot (a^{-5})^3
\][/tex]
3. Calculate the power of 6:
[tex]\[
6^3 = 6 \cdot 6 \cdot 6 = 216
\][/tex]
4. Apply the power rule to the variable term [tex]\((a^{-5})^3\)[/tex]:
[tex]\[
(a^{-5})^3 = a^{-5 \cdot 3} = a^{-15}
\][/tex]
5. Combine the results:
[tex]\[
6^3 \cdot (a^{-5})^3 = 216 \cdot a^{-15}
\][/tex]
6. Express the term with a positive exponent by using the reciprocal property [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]:
[tex]\[
216 \cdot a^{-15} = \frac{216}{a^{15}}
\][/tex]
Thus, the simplified expression [tex]\(\left(6 a^{-5}\right)^3\)[/tex] with positive exponents only is:
[tex]\[
\frac{216}{a^{15}}
\][/tex]
