To simplify the given expression:
[tex]\[
\frac{1}{2}+\left(\frac{3}{8}-\frac{1}{8}\right)^2
\][/tex]
we will proceed step by step.
### Step 1: Evaluate the expression inside the parentheses
First, evaluate [tex]\(\frac{3}{8} - \frac{1}{8}\)[/tex]:
[tex]\[
\frac{3}{8} - \frac{1}{8} = \frac{3 - 1}{8} = \frac{2}{8}
\][/tex]
Simplify [tex]\(\frac{2}{8}\)[/tex]:
[tex]\[
\frac{2}{8} = \frac{1}{4}
\][/tex]
So, we have:
[tex]\[
\left(\frac{3}{8} - \frac{1}{8}\right) = \frac{1}{4}
\][/tex]
### Step 2: Square the result from Step 1
Next, square [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\left(\frac{1}{4}\right)^2 = \frac{1^2}{4^2} = \frac{1}{16}
\][/tex]
### Step 3: Add [tex]\(\frac{1}{2}\)[/tex] to the squared result
Now, add [tex]\(\frac{1}{2}\)[/tex] to [tex]\(\frac{1}{16}\)[/tex]:
[tex]\[
\frac{1}{2} + \frac{1}{16}
\][/tex]
To add these fractions, find a common denominator. The least common multiple of 2 and 16 is 16. Convert [tex]\(\frac{1}{2}\)[/tex] to a fraction with denominator 16:
[tex]\[
\frac{1}{2} = \frac{1 \cdot 8}{2 \cdot 8} = \frac{8}{16}
\][/tex]
Add [tex]\(\frac{8}{16}\)[/tex] and [tex]\(\frac{1}{16}\)[/tex]:
[tex]\[
\frac{8}{16} + \frac{1}{16} = \frac{8 + 1}{16} = \frac{9}{16}
\][/tex]
### Final Answer
The simplified form of the given expression is:
[tex]\[
\boxed{\frac{9}{16}}
\][/tex]
