https://file.aiquickdraw.com/mathbot/file/17267474767109zgrxexm.jpg

Let's work through the problems step by step. I will start by explaining the solutions to each question:

### Question 9
Given \( a = \frac{1}{8}, b = 16, c = 4 \), the expression is \( a^{-\frac{1}{3}} \times b^{\frac{1}{4}} \times c^{\frac{1}{2}} \).

- \( a^{-\frac{1}{3}} = \left( \frac{1}{8} \right)^{-\frac{1}{3}} = 8^{\frac{1}{3}} = 2 \)
- \( b^{\frac{1}{4}} = 16^{\frac{1}{4}} = 2 \)
- \( c^{\frac{1}{2}} = 4^{\frac{1}{2}} = 2 \)

Now, multiply them:  
\( 2 \times 2 \times 2 = 8 \).

Thus, the answer is **not among the provided choices**. The result is 8, but none of the answers match this. There might be an error in the options.

### Question 10
Simplifying \( 2^{3x+2} \times 2^4 / 2^{2x+5} \).

- Using the law of exponents:  
  \( \frac{2^{3x+2} \times 2^4}{2^{2x+5}} = 2^{(3x+2) + 4 - (2x+5)} \)  
  \( = 2^{3x + 6 - 2x - 5} = 2^{x + 1} \)

Thus, the answer is \( \boxed{2^{x+1}} \), which corresponds to **Option A**.

### Question 11
Simplifying \( \left( \sqrt{5^3} \right)^6 \).

- \( \sqrt{5^3} = 5^{\frac{3}{2}} \)
- Now raise it to the 6th power:  
  \( \left( 5^{\frac{3}{2}} \right)^6 = 5^{\frac{3}{2} \times 6} = 5^9 \)

Thus, the answer is \( \boxed{5^9} \), but this option isn't listed either. There might be a mistake in the options again.

### Question 12
Simplifying \( (3\sqrt{5} - 1)^2 \).

- First, expand the square using the binomial formula:  
  \( (a - b)^2 = a^2 - 2ab + b^2 \)  
  Here, \( a = 3\sqrt{5}, b = 1 \).  
  \( = (3\sqrt{5})^2 - 2(3\sqrt{5})(1) + 1^2 \)  
  \( = 9 \times 5 - 6\sqrt{5} + 1 \)  
  \( = 45 - 6\sqrt{5} + 1 = 46 - 6\sqrt{5} \)

Thus, the answer is \( \boxed{46 - 6\sqrt{5}} \), which corresponds to **Option D**.

### Question 13
Simplifying \( \frac{1}{2} \times \sqrt{36} \times \sqrt{18} \).

- \( \sqrt{36} = 6 \)
- \( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \)

Now, multiply them:  
\( \frac{1}{2} \times 6 \times 3\sqrt{2} = 9\sqrt{2} \).

Thus, the answer is \( \boxed{9\sqrt{2}} \), but this option is also missing from the list.

### Question 14
Simplifying \( \frac{3\sqrt{5} - 10}{\sqrt{5}} \).

- Split the fraction:  
  \( \frac{3\sqrt{5}}{\sqrt{5}} - \frac{10}{\sqrt{5}} \)
- Simplify each term:  
  \( 3 - \frac{10}{\sqrt{5}} \)

Now, rationalize \( \frac{10}{\sqrt{5}} \):  
  \( \frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5} \).

Thus, the expression simplifies to \( 3 - 2\sqrt{5} \).

The answer is \( \boxed{3 - 2\sqrt{5}} \), which corresponds to **Option C**.

### Question 15
Simplifying \( \frac{13}{5 + 2\sqrt{3}} \).

- Rationalize the denominator by multiplying both numerator and denominator by the conjugate of \( 5 + 2\sqrt{3} \), which is \( 5 - 2\sqrt{3} \):
  \( \frac{13}{5 + 2\sqrt{3}} \times \frac{5 - 2\sqrt{3}}{5 - 2\sqrt{3}} = \frac{13(5 - 2\sqrt{3})}{(5 + 2\sqrt{3})(5 - 2\sqrt{3})} \)
- Simplify the denominator:  
  \( (5 + 2\sqrt{3})(5 - 2\sqrt{3}) = 25 - 12 = 13 \)

Thus, the expression becomes:  
\( \frac{13(5 - 2\sqrt{3})}{13} = 5 - 2\sqrt{3} \).

The answer is \( \boxed{5 - 2\sqrt{3}} \), which corresponds to **Option B**.

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Would you like further details on any particular problem? Here are 5 related questions for additional practice:

1. Simplify \( 2^{x+2} \times 2^{x+4} / 2^{2x+3} \).
2. Expand and simplify \( (4\sqrt{2} - 1)^2 \).
3. Simplify \( \frac{15}{4 + \sqrt{2}} \).
4. Simplify \( \left( \sqrt{3^4} \right)^5 \).
5. Evaluate \( 2^{3x+1} \times 2^{4x+3} / 2^{5x+4} \).

**Tip:** When simplifying radicals or exponents, always check for common bases to combine them easily.

Solve the given math problems in the image. Examples include finding simplified forms of exponents, solving radical expressions, and rationalizing denominators.

This problem set involves simplifying expressions with exponents, radicals, and rationalizing denominators. Students will explore key concepts like exponent laws, radical simplification, and binomial expansion. Suitable for Grades 9-12 students.

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