To simplify [tex]\(\left(2^{-5} \times 2^8\right)^2\)[/tex], we will utilize the properties of exponents step by step:
### Step 1: Simplify the expression inside the parentheses
We start by simplifying the expression [tex]\(2^{-5} \times 2^8\)[/tex] which is inside the parentheses.
Using the property of exponents that states [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
[tex]\[
2^{-5} \times 2^8 = 2^{-5 + 8}
\][/tex]
### Step 2: Combine the exponents
Next, we perform the addition in the exponent:
[tex]\[
-5 + 8 = 3
\][/tex]
So, the expression simplifies to:
[tex]\[
2^3
\][/tex]
Now, our original problem [tex]\(\left(2^{-5} \times 2^8\right)^2\)[/tex] has been simplified to [tex]\((2^3)^2\)[/tex].
### Step 3: Raise the simplified expression to the power of 2
The next step is to raise [tex]\(2^3\)[/tex] to the power of 2.
Using the property of exponents [tex]\((a^m)^n = a^{m \times n}\)[/tex]:
[tex]\[
(2^3)^2 = 2^{3 \times 2}
\][/tex]
### Step 4: Multiply the exponents
Perform the multiplication in the exponent:
[tex]\[
3 \times 2 = 6
\][/tex]
Thus, [tex]\((2^3)^2\)[/tex] simplifies to:
[tex]\[
2^6
\][/tex]
So, the simplified form of [tex]\(\left(2^{-5} \times 2^8\right)^2\)[/tex] is [tex]\(\boxed{2^6}\)[/tex].
