Sure, let's simplify the given mathematical expression [tex]\[
2: 0 \text{ valon de } \left[4^7 : 4^{10} : 4\right]^2 : \left(4^5\right)^7 e:
\][/tex]
#### Step 1: Simplify the Expression Inside the Brackets
We need to first simplify the expression inside the brackets: [tex]\[
\left[4^7 : 4^{10} : 4\right]
\][/tex]
Note that the colon (:) typically represents division here.
This can be rewritten and simplified step-by-step as follows:
[tex]\[
[4^7 : 4^{10} : 4] \rightarrow \frac{4^7}{4^{10}} : 4 = \frac{4^7}{4^{10} \times 4} = \frac{4^7}{4^{11}} = 4^{7-11} = 4^{-4}
\][/tex]
So the value of [tex]\(\left[4^7 : 4^{10} : 4\right]\)[/tex] simplifies to [tex]\(4^{-4}\)[/tex].
This further simplifies to:
[tex]\[
4^{-4} = \frac{1}{4^4} = \frac{1}{256} = 0.00390625
\][/tex]
#### Step 2: Square the Result
Next, square the value obtained from the brackets:
[tex]\[
\left[4^{-4}\right]^2 = \left(\frac{1}{256}\right)^2 = \frac{1}{(256^2)} = \frac{1}{65536} = 1.52587890625 \times 10^{-5}
\][/tex]
#### Step 3: Simplify [tex]\(\left(4^5\right)^7\)[/tex]
Now we simplify:
[tex]\[
(4^5)^7 = 4^{5 \times 7} = 4^{35} = 1180591620717411303424
\][/tex]
#### Step 4: Combining the Results
Next, we combine all the steps together:
[tex]\[
\left[4^7 : 4^{10} : 4\right]^2 \times (4^5)^7 = 1.52587890625 \times 10^{-5} \times 4^{35}
\][/tex]
The result of this entire calculation is:
[tex]\[
1.8014398509481984 \times 10^{16}
\][/tex]
#### Answer
Therefore, the closest answer in the provided options is 16.
Thus, the correct answer is:
[tex]\[
\boxed{16}
\][/tex]
