Answer :

To solve the expression [tex]\(\left(2 \frac{1}{2}\right)^2 + (0.5)^2\)[/tex], let's break it down step-by-step.

1. Convert the mixed number to an improper fraction or a decimal:
The mixed number [tex]\(2 \frac{1}{2}\)[/tex] can be converted to a decimal. [tex]\(2 \frac{1}{2} = 2 + \frac{1}{2} = 2.5\)[/tex].

2. Square each number:

- Square [tex]\(2.5\)[/tex]:
[tex]\[ (2.5)^2 = 2.5 \times 2.5 = 6.25 \][/tex]

- Square [tex]\(0.5\)[/tex]:
[tex]\[ (0.5)^2 = 0.5 \times 0.5 = 0.25 \][/tex]

3. Add the squares together:
[tex]\[ 6.25 + 0.25 = 6.5 \][/tex]

Therefore, [tex]\(\left(2 \frac{1}{2}\right)^2 + (0.5)^2 = 6.5\)[/tex].

So, our detailed steps give us the result:

1. [tex]\((2.5)^2 = 6.25\)[/tex]
2. [tex]\((0.5)^2 = 0.25\)[/tex]
3. [tex]\(6.25 + 0.25 = 6.5\)[/tex]

Thus, the final answer is:
[tex]\[ 6.5 \][/tex]