Let's solve the given expression step-by-step.
We are asked to find the value of the expression:
[tex]\[ e_0\left[(3,5)^4 \cdot(3,5)^3\right]^2 \][/tex]
We can break this problem into more manageable parts. First, let's evaluate [tex]\((3,5)^4 \cdot (3,5)^3\)[/tex].
Since [tex]\((3,5)\)[/tex] is a tuple representing the base of each element, we'll compute the power of each component separately and then multiply them together.
1. First, calculate [tex]\((3,5)^4\)[/tex]:
[tex]\[
(3,5)^4 = (3^4, 5^4)
\][/tex]
Evaluate each term:
[tex]\[
3^4 = 81 \quad \text{and} \quad 5^4 = 625
\][/tex]
So,
[tex]\[
(3,5)^4 = (81, 625)
\][/tex]
2. Next, calculate [tex]\((3,5)^3\)[/tex]:
[tex]\[
(3,5)^3 = (3^3, 5^3)
\][/tex]
Evaluate each term:
[tex]\[
3^3 = 27 \quad \text{and} \quad 5^3 = 125
\][/tex]
So,
[tex]\[
(3,5)^3 = (27, 125)
\][/tex]
3. Multiply the corresponding components of [tex]\((3,5)^4\)[/tex] and [tex]\((3,5)^3\)[/tex]:
[tex]\[
(3^4 \cdot 3^3, 5^4 \cdot 5^3) = (81 \cdot 27, 625 \cdot 125)
\][/tex]
Evaluate each term:
[tex]\[
81 \cdot 27 = 2187 \quad \text{and} \quad 625 \cdot 125 = 78125
\][/tex]
So,
[tex]\[
(3,5)^4 \cdot (3,5)^3 = (2187, 78125)
\][/tex]
4. Combine the results to get a single number:
Calculate the product of these two components:
[tex]\[
2187 \cdot 78125 = 170859375
\][/tex]
5. Raise this result to the power of 2:
[tex]\[
(170859375)^2
\][/tex]
6. Evaluate the final expression:
[tex]\[
170859375^2 = 29192926025390625
\][/tex]
Thus, the final result, step-by-step, is:
[tex]\[
\left[(3,5)^4 \cdot (3,5)^3\right]^2 = 29192926025390625
\][/tex]
