Let's analyze the given expression step by step:

$\frac{7 \sqrt{x^8 y}}{4} - \frac{7 \sqrt{xy}}{3x}$

Step 1: Simplify each square root term

First term: $\sqrt{x^8 y}$

• The square root of $x^8$ is $x^4$ because $\sqrt{x^8} = x^4$.
• So, $\sqrt{x^8 y} = x^4 \sqrt{y}$.

This gives the first term as: $\frac{7 x^4 \sqrt{y}}{4}$

Second term: $\sqrt{xy}$

• This term remains $\sqrt{xy}$ since there are no exponents higher than 2 under the square root.

This gives the second term as: $\frac{7 \sqrt{xy}}{3x}$

Step 2: Write the simplified expression

The entire expression now becomes: $\frac{7 x^4 \sqrt{y}}{4} - \frac{7 \sqrt{xy}}{3x}$

This is the simplified form of the given expression.

Would you like further details on any part of this solution or clarification?

Here are five related questions to further explore this concept:

1. How can we simplify expressions involving higher roots (e.g., cube roots)?
2. How does the assumption of positive real numbers affect square root simplifications?
3. Can you expand on rationalizing denominators when roots are involved?
4. What happens when terms under the square root have negative exponents?
5. How would you handle this problem if the expression had cube roots instead?

Tip: When simplifying square roots, always try to extract the largest possible perfect squares to simplify the expression.