Let's solve the given mathematical expression step by step:
Given expression: [tex]\(\left(2^5 \div 2^8\right)^5 \times 2^5\)[/tex]
1. Simplify the Division Inside the Parentheses:
The expression inside the parentheses is [tex]\(2^5 \div 2^8\)[/tex].
Using the properties of exponents, this can be simplified as:
[tex]\[
2^5 \div 2^8 = 2^{5-8} = 2^{-3}
\][/tex]
2. Apply the Exponent Outside the Parentheses:
Now, we need to raise [tex]\(2^{-3}\)[/tex] to the power of 5:
[tex]\[
\left(2^{-3}\right)^5 = 2^{-3 \times 5} = 2^{-15}
\][/tex]
3. Multiply by [tex]\(2^5\)[/tex]:
Next, we multiply [tex]\(2^{-15}\)[/tex] by [tex]\(2^5\)[/tex]:
[tex]\[
2^{-15} \times 2^5 = 2^{-15 + 5} = 2^{-10}
\][/tex]
4. Evaluate [tex]\(2^{-10}\)[/tex]:
The expression [tex]\(2^{-10}\)[/tex] is equivalent to:
[tex]\[
2^{-10} = \frac{1}{2^{10}} = \frac{1}{1024}
\][/tex]
So, the simplified result is [tex]\(2^{-10}\)[/tex], which is equal to [tex]\(\frac{1}{1024}\)[/tex].
Therefore, the correct answer is:
c. both [tex]\(a\)[/tex] and [tex]\(b\)[/tex]
