Let’s solve each of the expressions from a) to e) in simplest radical form:

a) $\sqrt{3a^2b} \times \sqrt{6ab^5}$

We combine the square roots by multiplying the terms inside: $\sqrt{3a^2b} \times \sqrt{6ab^5} = \sqrt{(3a^2b) \times (6ab^5)} = \sqrt{18a^3b^6}$ Now simplify:

• The square root of $a^3$ is $a$ (because $\sqrt{a^2} = a$ and $a^3 = a^2 \times a$, so we pull out $a$).
• The square root of $b^6$ is $b^3$ (since $\sqrt{b^6} = b^3$).

Thus, the simplified expression is: $a b^3 \sqrt{18a}$ We can simplify the radical further: $\sqrt{18a} = \sqrt{9 \times 2a} = 3\sqrt{2a}$ Final answer: $3ab^3\sqrt{2a}$

b) $\sqrt{6ab^5}$

As already shown, we can simplify this by breaking it into parts:

• The square root of $b^5$ is $b^2 \times \sqrt{b}$. So, the expression becomes: $b^2 \sqrt{6ab}$

c) $(2x^3 \sqrt{2y^4}) \times (x^2 \sqrt{4y^2})$

First, multiply the constants and powers of $x$ and $y$:

• $2x^3 \times x^2 = 2x^5$
• $\sqrt{2y^4} \times \sqrt{4y^2} = \sqrt{(2y^4) \times (4y^2)} = \sqrt{8y^6}$

Now simplify the square root: $\sqrt{8y^6} = \sqrt{4 \times 2 \times y^6} = 2y^3 \sqrt{2}$

Final answer: $2x^5y^3 \sqrt{2}$

d) $\frac{9x^2 \sqrt{x^2y^5}}{3x^5 \sqrt{x^6y}}$

First, simplify the coefficients and powers of $x$ and $y$:

• The coefficient is $\frac{9}{3} = 3$
• For $x$, the powers become: $\frac{x^2}{x^5} = x^{-3} = \frac{1}{x^3}$
• For $y$, simplify the square roots: $\frac{\sqrt{x^2y^5}}{\sqrt{x^6y}} = \sqrt{\frac{x^2y^5}{x^6y}} = \sqrt{\frac{y^4}{x^4}} = \frac{y^2}{x^2}$

Thus, the final expression is: $\frac{3y^2}{x^5}$

e) $\frac{9x^2 \sqrt{x^2y^5}}{3x^5 \sqrt{x^6y}}$

We already solved this in d), so the final simplified form is: $\frac{3y^2}{x^5}$

Summary of Answers:

• a) $3ab^3\sqrt{2a}$
• b) $b^2\sqrt{6ab}$
• c) $2x^5y^3 \sqrt{2}$
• d) $\frac{3y^2}{x^5}$
• e) $\frac{3y^2}{x^5}$

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

Here are some related questions:

1. How do you simplify a square root when there are variables involved?
2. How do you combine square roots with different bases (e.g., $\sqrt{a} \times \sqrt{b}$)?
3. Can you apply the same method to cube roots or higher roots? How?
4. What is the importance of simplifying radicals in algebraic expressions?
5. How do you approach simplifying fractions with radicals in the numerator and denominator?

Tip: When simplifying square roots, always look for perfect squares (or higher powers) inside the root to make simplification easier!