Let's solve each problem on this worksheet.

1. $(-7)^2$

$(-7)^2 = (-7) \times (-7) = 49$

2. $- (7^2)$

$7^2 = 49, \quad \text{so } - (7^2) = -49$

3. $- \left(\frac{1}{3}\right)^2$

$\left(\frac{1}{3}\right)^2 = \frac{1}{9}, \quad \text{so } - \frac{1}{9} = -\frac{1}{9}$

4. $\left(-\frac{1}{5}\right)^2$

$\left(-\frac{1}{5}\right)^2 = \frac{1}{25}$

5. $-1.3(4) + 7$

$-1.3 \times 4 = -5.2, \quad -5.2 + 7 = 1.8$

6. $12 \left(\frac{1}{4}\right) - 12 \left(-\frac{2}{3}\right)$

$12 \times \frac{1}{4} = 3 \quad \text{and} \quad 12 \times -\frac{2}{3} = -8$ $3 - (-8) = 3 + 8 = 11$

7. $10 - \frac{5}{6}(-4) + 2$

$\frac{5}{6} \times -4 = -\frac{20}{6} = -\frac{10}{3}$ $10 - (-\frac{10}{3}) + 2 = 10 + \frac{10}{3} + 2 = \frac{30}{3} + \frac{10}{3} + \frac{6}{3} = \frac{46}{3}$

8. $8 - (4^3)$

$4^3 = 64, \quad \text{so } 8 - 64 = -56$

If you would like a detailed breakdown of any specific problem or have additional questions, feel free to ask.

Here are five questions related to the concepts used:

1. What is the difference between squaring a negative number and the negative of a square?
2. How do fractional exponents impact the value of an expression?
3. What is the role of parentheses in calculations involving negative numbers?
4. How do you handle addition and subtraction when working with mixed fractions?
5. How can you simplify complex fractions before calculating?

Tip: When evaluating expressions, always pay close attention to the parentheses as they dictate the order of operations and can change the result significantly.