https://file.aiquickdraw.com/mathbot/file/1725505095315217r0lqk.jpg

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!

Simplify the following radicals:
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. \( \frac{\sqrt[5]{625m^6n^{10}}}{\sqrt[7]{27m^{21}}} \cdot 4m^3n^4 \)

Learn how to simplify complex radical expressions involving cube roots, fourth roots, and fractional exponents. This example covers properties of radicals, exponent rules, and algebraic simplifications, ideal for high school students in Grades 10-12.

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