Let's go through the problems step by step.

Problem (a):

$\sqrt[3]{\frac{27x^9y^{18}}{2x^{10}y^8}}$

Step 1: Simplify the expression inside the cube root.

• Numerator: $27x^9y^{18}$
• Denominator: $2x^{10}y^8$

When simplifying, subtract the exponents for like bases:

$\frac{27x^9y^{18}}{2x^{10}y^8} = \frac{27}{2} \times \frac{x^9}{x^{10}} \times \frac{y^{18}}{y^8} = \frac{27}{2} \times x^{-1} \times y^{10}$

So the expression inside the cube root becomes:

$\sqrt[3]{\frac{27}{2} \times x^{-1} \times y^{10}}$

Step 2: Apply the cube root.

Distribute the cube root over each term:

[ \sqrt[3]{\frac{27}{2}} \times \sqrt[3]{x^{-1}} \times \sqrt[3]{y^{10}} ]

Simplify each cube root:

[ \sqrt[3]{27} = 3, \quad \sqrt[3]{x^{-1}} = x^{-\frac{1}{3}}, \quad \text{and} \quad \sqrt[3]{y^{10}} = y^{\frac{10}{3}} ]

So the final expression is:

$\frac{3y^{\frac{10}{3}}}{\sqrt[3]{2} \times x^{\frac{1}{3}}}$

Problem (b):

$\sqrt{\frac{3}{5} \times \frac{x^2 \cdot 2x^4}{4x^6}}$

Step 1: Simplify the expression inside the square root.

First, simplify inside the square root:

$\frac{3}{5} \times \frac{x^2 \times 2x^4}{4x^6} = \frac{3}{5} \times \frac{2x^{6}}{4x^6} = \frac{3 \times 2}{5 \times 4} = \frac{6}{20} = \frac{3}{10}$

The expression inside the square root simplifies to $\frac{3}{10}$.

Step 2: Apply the square root.

Take the square root of each part:

$\sqrt{\frac{3}{10}} = \frac{\sqrt{3}}{\sqrt{10}} = \frac{\sqrt{3}}{10^{1/2}}$

This simplifies to:

$\frac{\sqrt{3}}{\sqrt{10}}$

To rationalize the denominator:

$\frac{\sqrt{3}}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{30}}{10}$

Thus, the simplified form is:

$\frac{\sqrt{30}}{10}$

Summary of Answers:

• (a): $\frac{3y^{\frac{10}{3}}}{\sqrt[3]{2} \times x^{\frac{1}{3}}}$
• (b): $\frac{\sqrt{30}}{10}$

Would you like further details or have any questions?

Additional Questions to Explore:

1. How do you simplify cube roots with fractional exponents?
2. What steps are involved in rationalizing a denominator?
3. How does simplifying within radicals differ when dealing with cube roots versus square roots?
4. What are the general rules for simplifying expressions with exponents in fractions?
5. How do you approach simplifying a radical expression involving both numbers and variables?

Tip:

When simplifying radicals, always check if the denominator can be rationalized to make the expression more manageable.