Let's evaluate each expression step-by-step, using the order of operations (PEMDAS/BODMAS).
### Expression 1:
[tex]\[
(2 - 4)^2 - 7
\][/tex]
1. First, subtract inside the parentheses:
[tex]\[
2 - 4 = -2
\][/tex]
2. Then, square the result:
[tex]\[
(-2)^2 = 4
\][/tex]
3. Finally, subtract 7:
[tex]\[
4 - 7 = -3
\][/tex]
So,
[tex]\[
(2 - 4)^2 - 7 = -3
\][/tex]
### Expression 2:
[tex]\[
(-7)^2 - 7(-7) - 9 \cdot 9^2 + 5
\][/tex]
1. First, evaluate the exponentiation:
[tex]\[
(-7)^2 = 49
\][/tex]
[tex]\[
9^2 = 81
\][/tex]
2. Then, multiply:
[tex]\[
7(-7) = -49
\][/tex]
[tex]\[
9 \cdot 81 = 729
\][/tex]
3. Substitute these values back into the expression:
[tex]\[
49 - (-49) - 729 + 5
\][/tex]
4. Simplify within the subtractions and additions:
- Change the minus a negative to a plus:
[tex]\[
49 + 49 - 729 + 5
\][/tex]
5. Finally, add and subtract in order from left to right:
[tex]\[
98 - 729 + 5 = -631 + 5 = -626
\][/tex]
So,
[tex]\[
(-7)^2 - 7(-7) - 9 \cdot 9^2 + 5 = -626
\][/tex]
### Expression 3:
[tex]\[
7 - 6(9^2 - 5) - (-2)
\][/tex]
1. First, evaluate inside the parentheses:
[tex]\[
9^2 = 81
\][/tex]
[tex]\[
81 - 5 = 76
\][/tex]
2. Substitute back into the expression:
[tex]\[
7 - 6(76) - (-2)
\][/tex]
3. Then, multiply:
[tex]\[
6 \cdot 76 = 456
\][/tex]
4. Substitute and simplify:
[tex]\[
7 - 456 + 2
\][/tex]
5. Finally, perform the additions and subtractions in order:
[tex]\[
7 - 456 + 2 = -449 + 2 = -447
\][/tex]
So,
[tex]\[
7 - 6(9^2 - 5) - (-2) = -447
\][/tex]
### Expression 4:
[tex]\[
5(-3)^2 + 63 \div (-9)
\][/tex]
1. First, evaluate the exponentiation:
[tex]\[
(-3)^2 = 9
\][/tex]
2. Substitute back into the expression and multiply:
[tex]\[
5 \cdot 9 + 63 \div (-9)
\][/tex]
3. Then, perform the division:
[tex]\[
63 \div (-9) = -7
\][/tex]
4. Finally, add the results:
[tex]\[
45 - 7 = 38
\][/tex]
So,
[tex]\[
5(-3)^2 + 63 \div (-9) = 38
\][/tex]
### Expression 5:
[tex]\[
(-5) - (-4)(2^3 - (-3))
\][/tex]
1. First, evaluate the exponentiation:
[tex]\[
2^3 = 8
\][/tex]
2. Then, evaluate inside the inner parentheses:
[tex]\[
8 - (-3) = 8 + 3 = 11
\][/tex]
3. Substitute back into the expression and multiply:
[tex]\[
(-5) - (-4)(11)
\][/tex]
[tex]\[
(-4) \cdot 11 = -44
\][/tex]
4. Finally, perform the subtraction:
[tex]\[
(-5) + 44 = 39
\][/tex]
So,
[tex]\[
(-5) - (-4)(2^3 - (-3)) = 39
\][/tex]
The evaluated results are:
[tex]\[
\begin{array}{c|c}
(2-4)^2-7 = -3 \\
(-7)^2 - 7(-7) - 9 \cdot 9^2 + 5 = -626 \\
7 - 6(9^2 - 5) - (-2) = -447 & 5(-3)^2 + 63 \div (-9) = 38 \\
& (-5) - (-4)(2^3 - (-3)) = 39
\end{array}
\][/tex]
