To simplify the given expression [tex]\(\left(9 - \frac{2}{3} \sqrt{9}\right)^2 + (1 - 6)^2\)[/tex], let's break it down step by step:
1. Simplify inside the parentheses:
- Start with the term [tex]\(\sqrt{9}\)[/tex]. The square root of 9 is 3:
[tex]\[
\sqrt{9} = 3
\][/tex]
- Next, substitute [tex]\(\sqrt{9}\)[/tex] into the expression:
[tex]\[
\frac{2}{3} \sqrt{9} = \frac{2}{3} \times 3
\][/tex]
- Multiplying the fraction by 3:
[tex]\[
\frac{2}{3} \times 3 = 2
\][/tex]
- Now substitute back into the expression:
[tex]\[
9 - 2 = 7
\][/tex]
2. Square the first term:
- After simplifying inside the parentheses, we have:
[tex]\[
\left( 9 - \frac{2}{3} \times 3 \right)^2 = 7^2
\][/tex]
- Now calculate the square of 7:
[tex]\[
7^2 = 49
\][/tex]
3. Simplify the second term:
- Inside the parentheses:
[tex]\[
1 - 6 = -5
\][/tex]
- Square this result:
[tex]\[
(-5)^2 = 25
\][/tex]
4. Add the squared terms:
- Now add the two squared results together:
[tex]\[
49 + 25 = 74
\][/tex]
Therefore, the simplified expression evaluates to:
[tex]\[
\boxed{74}
\][/tex]
