Sure, let's solve the given expression step-by-step:
Given expression:
[tex]\[
\left[\left(\frac{2}{13}\right)^{-6} \div \left(\frac{2}{13}\right)^3\right]^3 \times \left(\frac{2}{13}\right)^{-9}
\][/tex]
Step 1: Simplify the inner division.
Recall the rule of exponents:
[tex]\[
a^{-m} \div a^n = a^{-m - n}
\][/tex]
Applying this rule, we get:
[tex]\[
\left(\frac{2}{13}\right)^{-6} \div \left(\frac{2}{13}\right)^3 = \left(\frac{2}{13}\right)^{-6 - 3} = \left(\frac{2}{13}\right)^{-9}
\][/tex]
Step 2: Raise the result to the power of 3.
Using the power of a power rule:
[tex]\[
(a^m)^n = a^{m \cdot n}
\][/tex]
We have:
[tex]\[
\left[\left(\frac{2}{13}\right)^{-9}\right]^3 = \left(\frac{2}{13}\right)^{-9 \cdot 3} = \left(\frac{2}{13}\right)^{-27}
\][/tex]
Step 3: Multiply with [tex]\(\left(\frac{2}{13}\right)^{-9}\)[/tex].
Using the product of powers rule:
[tex]\[
a^m \times a^n = a^{m + n}
\][/tex]
We get:
[tex]\[
\left(\frac{2}{13}\right)^{-27} \times \left(\frac{2}{13}\right)^{-9} = \left(\frac{2}{13}\right)^{-27 - 9} = \left(\frac{2}{13}\right)^{-36}
\][/tex]
Now, [tex]\(\left(\frac{2}{13}\right)^{-36}\)[/tex] can be simplified further using the definition of negative exponents:
[tex]\[
a^{-m} = \frac{1}{a^m}
\][/tex]
Thus:
[tex]\[
\left(\frac{2}{13}\right)^{-36} = \left(\frac{13}{2}\right)^{36}
\][/tex]
Finally, evaluating this expression numerically:
[tex]\[
\left(\frac{13}{2}\right)^{36} \approx 1.8402670033873177 \times 10^{29}
\][/tex]
Therefore, the final result is approximately:
[tex]\[
1.8402670033873177 \times 10^{29}
\][/tex]
