To simplify the expression [tex]\( 8 + \left[4^2 - 6 + (5 + 1)\right] \)[/tex] using the order of operations, we follow these steps:
1. Parentheses and Brackets: Solve the innermost parentheses first.
[tex]\[
5 + 1 = 6
\][/tex]
So the expression becomes:
[tex]\[
8 + \left[4^2 - 6 + 6\right]
\][/tex]
2. Exponents: Next, we handle the exponentiation.
[tex]\[
4^2 = 16
\][/tex]
So the expression now is:
[tex]\[
8 + \left[16 - 6 + 6\right]
\][/tex]
3. Addition/Subtraction within Brackets: Evaluate the operations inside the brackets from left to right.
[tex]\[
16 - 6 = 10
\][/tex]
Then:
[tex]\[
10 + 6 = 16
\][/tex]
So the expression reduces to:
[tex]\[
8 + 16
\][/tex]
4. Final Addition: Lastly, we add the remaining values.
[tex]\[
8 + 16 = 24
\][/tex]
Therefore, the simplified value of the expression [tex]\( 8 + \left[4^2 - 6 + (5 + 1)\right] \)[/tex] is [tex]\(\boxed{24}\)[/tex].
