https://file.aiquickdraw.com/mathbot/file/17265970359630qbgh5im.jpg

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**

---

### **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) \(\frac{26}{3}\)**

---

### **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) \(- \frac{55}{4}\)**

---

### **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) \(\frac{17}{5}\)**

---

### **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**

---

### **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) \(- \frac{5}{13}\)**

---

### **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.

This page provides solutions to a variety of math problems involving exponents, fractional powers, and arithmetic operations. Suitable for students in Grades 9-10, these problems help practice key algebraic concepts.

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