To simplify the given expression [tex]\(\left(7 - \frac{1}{4} \sqrt{16}\right)^2 + (2 - 5)^2\)[/tex], follow these steps:
1. Simplify the square root and fractions inside the parentheses:
- Calculate the square root [tex]\(\sqrt{16}\)[/tex]. Since [tex]\(\sqrt{16} = 4\)[/tex],
[tex]\[
\left(7 - \frac{1}{4} \times 4\right)^2
\][/tex]
- Simplify the fraction [tex]\(\frac{1}{4} \times 4 = 1\)[/tex].
2. Subtract inside the first parentheses:
- Now we have:
[tex]\[
7 - 1 = 6
\][/tex]
- So the first part simplifies to:
[tex]\[
\left(6\right)^2
\][/tex]
3. Simplify the second part inside the parentheses:
[tex]\[
2 - 5 = -3
\][/tex]
- So the second part simplifies to:
[tex]\[
\left(-3\right)^2
\][/tex]
4. Calculate the squares of the simplified expressions:
- [tex]\((6)^2 = 36\)[/tex]
- [tex]\((-3)^2 = 9\)[/tex]
5. Add the squared values together:
[tex]\[
36 + 9 = 45
\][/tex]
So, the simplified result of the expression [tex]\(\left(7 - \frac{1}{4} \sqrt{16}\right)^2 + (2 - 5)^2\)[/tex] is [tex]\(\boxed{45}\)[/tex].
