Let's go through each problem step by step and solve the given expressions.

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Question 9:

Given $a = 3$, find the value of $a^2 + a^3$.

$a^2 + a^3 = 3^2 + 3^3 = 9 + 27 = 36$ Answer: A) 36

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Question 10:

Given $x = -2$, find the value of $x^t$.

The expression given is $x^t$, but it is not clear what $t$ represents here. Assuming $t = 2$, we have $x^2$.

$(-2)^2 = 4$ Answer: E) 4

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Question 11:

Solve $4^1 + 5^1 - 3^{-1}$.

First, calculate the powers: $4^1 = 4, \quad 5^1 = 5, \quad 3^{-1} = \frac{1}{3}$ Now, solve the expression: $4 + 5 - \frac{1}{3} = 9 - \frac{1}{3} = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}$ Answer: C) $\frac{26}{3}$

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Question 12:

Solve $\left( \frac{3}{2} \right)^2 - \left( \frac{1}{4} \right)^{-2}$.

First, calculate each power: $\left( \frac{3}{2} \right)^2 = \frac{9}{4}, \quad \left( \frac{1}{4} \right)^{-2} = 4^2 = 16$ Now, subtract: $\frac{9}{4} - 16 = \frac{9}{4} - \frac{64}{4} = \frac{9 - 64}{4} = \frac{-55}{4}$ Answer: B) $- \frac{55}{4}$

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Question 13:

Solve $\frac{5^2 + (-2)^3}{(-1)^6 + 2^2}$.

First, calculate the powers: $5^2 = 25, \quad (-2)^3 = -8, \quad (-1)^6 = 1, \quad 2^2 = 4$ Now, solve: $\frac{25 + (-8)}{1 + 4} = \frac{17}{5}$ Answer: A) $\frac{17}{5}$

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Question 14:

Solve $\frac{(3^1 + 3^0)^2 \cdot 3^{-3}}{3^{-3}}$.

First, calculate the powers: $3^1 = 3, \quad 3^0 = 1 \quad \Rightarrow \quad (3 + 1)^2 = 4^2 = 16$ Now simplify: $\frac{16 \cdot 3^{-3}}{3^{-3}} = 16$ Answer: A) 144

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Question 15:

Solve $\frac{-2^3 + (3^{-1})^{-1}}{2^2 + 3^2}$.

First, calculate the powers: $-2^3 = -8, \quad (3^{-1})^{-1} = 3, \quad 2^2 = 4, \quad 3^2 = 9$ Now, solve: $\frac{-8 + 3}{4 + 9} = \frac{-5}{13}$ Answer: B) $- \frac{5}{13}$

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Question 16:

Solve $\frac{3^{-1} - 5^{-1}}{3^1 + 5^1}$.

First, calculate the powers: $3^{-1} = \frac{1}{3}, \quad 5^{-1} = \frac{1}{5}, \quad 3^1 = 3, \quad 5^1 = 5$ Now, solve the numerator and denominator: $\frac{1}{3} - \frac{1}{5} = \frac{5 - 3}{15} = \frac{2}{15}, \quad 3 + 5 = 8$ So: $\frac{\frac{2}{15}}{8} = \frac{2}{15} \times \frac{1}{8} = \frac{2}{120} = \frac{1}{60}$ Answer: D) $\frac{1}{60}$

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Do you have any questions or need further details on any of these problems?

Here are some related questions to consider:

1. How can we generalize exponent rules for fractional exponents?
2. What are the differences between positive and negative exponents?
3. How does raising a fraction to a negative power work?
4. Can powers of negative numbers be simplified similarly to powers of positive numbers?
5. How does the order of operations affect expressions with multiple exponents?

Tip: Always simplify powers and roots before tackling addition or subtraction when dealing with complex expressions.