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.

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.