To simplify the given expression, we can break it down step-by-step.
Given expression:
[tex]\[
\frac{1}{2} + \left(\frac{3}{8} - \frac{1}{8}\right)^2
\][/tex]
Step 1: Simplify the expression inside the parentheses first:
[tex]\[
\frac{3}{8} - \frac{1}{8}
\][/tex]
Since the denominators are the same, we can simply subtract the numerators:
[tex]\[
\frac{3 - 1}{8} = \frac{2}{8}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{2}{8} = \frac{1}{4}
\][/tex]
Step 2: Square the result of the simplified fraction inside the parentheses:
[tex]\[
\left(\frac{1}{4}\right)^2 = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}
\][/tex]
Step 3: Add this result to [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{2} + \frac{1}{16}
\][/tex]
To add these fractions, we need a common denominator. The common denominator of 2 and 16 is 16. Convert [tex]\(\frac{1}{2}\)[/tex] to a fraction with the denominator 16:
[tex]\[
\frac{1}{2} = \frac{8}{16}
\][/tex]
Now add:
[tex]\[
\frac{8}{16} + \frac{1}{16} = \frac{8+1}{16} = \frac{9}{16}
\][/tex]
Thus, the simplified form of the expression [tex]\(\frac{1}{2} + \left(\frac{3}{8} - \frac{1}{8}\right)^2\)[/tex] is:
[tex]\[
\boxed{\frac{9}{16}}
\][/tex]
