Certainly! Let's evaluate each expression step-by-step by following the order of operations (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
1. Evaluating [tex]\((8-6)^2 - 3\)[/tex]:
[tex]\[
(8-6) = 2 \quad \text{(perform the subtraction inside the parentheses)}
\][/tex]
[tex]\[
2^2 = 4 \quad \text{(square the result)}
\][/tex]
[tex]\[
4 - 3 = 1 \quad \text{(subtract)}
\][/tex]
So, [tex]\((8-6)^2 - 3 = 1\)[/tex].
2. Evaluating [tex]\(9 \cdot 7^2 + 2\)[/tex]:
[tex]\[
7^2 = 49 \quad \text{(square \(7\))}
\][/tex]
[tex]\[
9 \cdot 49 = 441 \quad \text{(multiply)}
\][/tex]
[tex]\[
441 + 2 = 443 \quad \text{(add)}
\][/tex]
So, [tex]\(9 \cdot 7^2 + 2 = 443\)[/tex].
3. Evaluating [tex]\((-5)^2 - 3(-5) - 7\)[/tex]:
[tex]\[
(-5)^2 = 25 \quad \text{(square \(-5\))}
\][/tex]
[tex]\[
-3(-5) = 15 \quad \text{(multiply \( -3 \) by \(-5\))}
\][/tex]
[tex]\[
25 + 15 = 40 \quad \text{(add)}
\][/tex]
[tex]\[
40 - 7 = 33 \quad \text{(subtract)}
\][/tex]
So, [tex]\((-5)^2 - 3(-5) - 7 = 33\)[/tex].
4. Evaluating [tex]\(3 - 9(7^2 - 2) - (-2)\)[/tex]:
[tex]\[
7^2 = 49 \quad \text{(square \(7\))}
\][/tex]
[tex]\[
49 - 2 = 47 \quad \text{(subtract inside the parentheses)}
\][/tex]
[tex]\[
9 \cdot 47 = 423 \quad \text{(multiply)}
\][/tex]
[tex]\[
3 - 423 = -420 \quad \text{(subtract)}
\][/tex]
[tex]\[
-420 + 2 = -418 \quad \text{(adding \(-(-2)\))}
\][/tex]
So, [tex]\(3 - 9(7^2 - 2) - (-2) = -418\)[/tex].
5. Evaluating [tex]\(2(-3)^2 + 30 \div (-6)\)[/tex]:
[tex]\[
(-3)^2 = 9 \quad \text{(square \(-3\))}
\][/tex]
[tex]\[
2 \cdot 9 = 18 \quad \text{(multiply)}
\][/tex]
[tex]\[
30 \div (-6) = -5 \quad \text{(divide)}
\][/tex]
[tex]\[
18 + (-5) = 13 \quad \text{(add)}
\][/tex]
So, [tex]\(2(-3)^2 + 30 \div (-6) = 13.0\)[/tex].
6. Evaluating [tex]\((-9) - (-7)(2^3 - (-8))\)[/tex]:
[tex]\[
2^3 = 8 \quad \text{(cube \(2\))}
\][/tex]
[tex]\[
-8 + 8 = 16 \quad \text{(subtract inside the parentheses)}
\][/tex]
[tex]\[
-7 \cdot 16 = -112 \quad \text{(multiply)}
\][/tex]
[tex]\[
-9 - (-112) = -9 + 112 = 103 \quad \text{(subtract by adding the positive value)}
\][/tex]
So, [tex]\((-9) - (-7)(2^3 - (-8)) = 103\)[/tex].
Thus, the final answers are:
1. [tex]\( (8-6)^2 - 3 = 1 \)[/tex]
2. [tex]\( 9 \cdot 7^2 + 2 = 443 \)[/tex]
3. [tex]\( (-5)^2 - 3(-5) - 7 = 33 \)[/tex]
4. [tex]\( 3 - 9(7^2 - 2) - (-2) = -418 \)[/tex]
5. [tex]\( 2(-3)^2 + 30 \div (-6) = 13.0 \)[/tex]
6. [tex]\( (-9) - (-7)(2^3 - (-8)) = 103 \)[/tex]
