Let's go through the problems you uploaded one by one:

Problem 4:

You are asked to simplify the expression:

$\frac{24a^7b^{-2}c}{6a^{-2}b^{-3}c^{-6}}$

First, simplify the coefficients and variables separately:

• Coefficients: $\frac{24}{6} = 4$
• For $a$, subtract the exponents: $a^{7} \div a^{-2} = a^{7 - (-2)} = a^9$
• For $b$, subtract the exponents: $b^{-2} \div b^{-3} = b^{-2 - (-3)} = b^1$
• For $c$, subtract the exponents: $c^{1} \div c^{-6} = c^{1 - (-6)} = c^7$

The simplified form is:

$4a^9bc^7$

From the options, the correct answer is not listed directly, but we can match based on this expression. The closest match is likely option d, which aligns with the structure.

Problem 5:

You need to simplify the expression:

$\frac{3}{2 - \sqrt{5}}$

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:

$\frac{3}{2 - \sqrt{5}} \times \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{3(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})}$

Now, calculate the denominator:

$(2 - \sqrt{5})(2 + \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1$

Thus, the expression becomes:

$\frac{3(2 + \sqrt{5})}{-1} = -3(2 + \sqrt{5}) = -6 - 3\sqrt{5}$

Now, compare this with $p + q\sqrt{r}$:

$p = -6, \quad q = -3, \quad r = 5$

The question asks for the value of $p - q + r$:

$p - q + r = -6 - (-3) + 5 = -6 + 3 + 5 = 2$

So, the answer is d. 2.

Problem 6:

You need to simplify:

$\sqrt{20} - 2\sqrt{45} + \sqrt{125} - \sqrt{80}$

First, simplify each square root:

• $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$
• $2\sqrt{45} = 2\sqrt{9 \times 5} = 6\sqrt{5}$
• $\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}$
• $\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$

Now, substitute back:

$2\sqrt{5} - 6\sqrt{5} + 5\sqrt{5} - 4\sqrt{5}$

Combine like terms:

$(2 - 6 + 5 - 4)\sqrt{5} = -3\sqrt{5}$

So, the answer is c. -3\sqrt{5}.

Let me know if you need more details or have any questions!

Related Questions:

1. How do you handle rationalizing denominators with cube roots?
2. What are the rules for simplifying exponents with different bases?
3. How do you solve equations involving radicals like $\sqrt{x}$ or cube roots?
4. Can you generalize the process for factoring expressions with square roots?
5. How do you simplify a complex fraction with multiple radicals?

Tip:

When rationalizing denominators, always multiply by the conjugate if you have a binomial with a square root in the denominator!