I have reviewed the image, and it contains five radical expressions to simplify. Let's go through them step by step:

1. $\sqrt[3]{\frac{27x^9y^3}{125x^6y^5}}$

2. $\sqrt[3]{64m^3n^9}$

3. $\sqrt[3]{20a^{15}b^{24}c^7}$

4. $\sqrt[4]{\frac{48x^{15}y^5z^3}{2050x^{18}}} \cdot 2\left(\sqrt[4]{\frac{16x^8}{y^{12}z^9}}\right)$

5. $\frac{\sqrt[5]{625m^6n^{10}}}{\sqrt[7]{27m^{21}}} \cdot 4m^3n^4$

Let me simplify each expression for you.

1. $\sqrt[3]{\frac{27x^9y^3}{125x^6y^5}}$

• Simplify the constants: $\frac{27}{125} = \left(\frac{3}{5}\right)^3$
• Exponents of $x$: $\frac{x^9}{x^6} = x^3$
• Exponents of $y$: $\frac{y^3}{y^5} = y^{-2}$

Now take the cube root: $\frac{3}{5} x y^{-\frac{2}{3}}$

2. $\sqrt[3]{64m^3n^9}$

• $64 = 4^3$
• Cube root of $m^3$ and $n^9$: $\sqrt[3]{64m^3n^9} = 4mn^3$

3. $\sqrt[3]{20a^{15}b^{24}c^7}$

• Split constants: $20 = 2^2 \times 5$
• Cube root of $a^{15} = a^5$ and $b^{24} = b^8$, while $c^7$ remains $c^{\frac{7}{3}}$.

Result: $2a^5b^8c^{\frac{7}{3}}$

4. $\sqrt[4]{\frac{48x^{15}y^5z^3}{2050x^{18}}} \cdot 2\left(\sqrt[4]{\frac{16x^8}{y^{12}z^9}}\right)$

Simplify the first term:

• Divide $x^{15} / x^{18} = x^{-3}$.
• Leave the rest: $\frac{48y^5z^3}{2050}$

Apply the 4th root:

• $48/2050 = \frac{24}{1025}$, which doesn't simplify much.
• Exponents: $y^{5/4} z^{3/4} \cdot x^{-3/4}$

Now, simplify the second term:

• $\sqrt[4]{16x^8} = 2x^2$
• $\sqrt[4]{y^{12}z^9} = y^3z^{9/4}$

Multiply both results.

5. $\frac{\sqrt[5]{625m^6n^{10}}}{\sqrt[7]{27m^{21}}} \cdot 4m^3n^4$

• $\sqrt[5]{625} = 5$, and $\sqrt[7]{27} = 3$.
• For $m$, simplify: $\frac{m^6}{m^{21/7}} = m^{6 - 3} = m^3$.
• For $n$, simplify: $n^2$.

Multiply by $4m^3n^4$: Result: $\frac{5m^6n^6}{3}$.

Do you want any steps clarified or expanded? Here are 5 related questions:

1. How do we handle fractional exponents when simplifying radicals?
2. How does the cube root affect powers of variables like $x^9$?
3. What happens when dividing exponents with the same base?
4. How do fourth roots work when multiplying radical expressions?
5. Can complex radicals be simplified using factorization?

Tip: Always factor constants and variables before applying roots; this simplifies the process!