To simplify the given expression, let's break down each term. The expression is:

$\sqrt{x^5 y^7} - \sqrt{x^7 y^5} + \sqrt{x^{13} y}$

Step 1: Simplify each square root

1. Simplify $\sqrt{x^5 y^7}$:

   $\sqrt{x^5 y^7} = \sqrt{x^4 \cdot x \cdot y^6 \cdot y} = \sqrt{x^4} \cdot \sqrt{x} \cdot \sqrt{y^6} \cdot \sqrt{y}$

   Simplifying further:

   $= x^2 \cdot \sqrt{x} \cdot y^3 \cdot \sqrt{y} = x^2 y^3 \sqrt{xy}$

2. Simplify $\sqrt{x^7 y^5}$:

   $\sqrt{x^7 y^5} = \sqrt{x^6 \cdot x \cdot y^4 \cdot y} = \sqrt{x^6} \cdot \sqrt{x} \cdot \sqrt{y^4} \cdot \sqrt{y}$

   Simplifying further:

   $= x^3 \cdot \sqrt{x} \cdot y^2 \cdot \sqrt{y} = x^3 y^2 \sqrt{xy}$

3. Simplify $\sqrt{x^{13} y}$:

   $\sqrt{x^{13} y} = \sqrt{x^{12} \cdot x \cdot y} = \sqrt{x^{12}} \cdot \sqrt{x} \cdot \sqrt{y}$

   Simplifying further:

   $= x^6 \cdot \sqrt{xy}$

Step 2: Combine the simplified terms

Now that each term is simplified, we combine them:

$x^2 y^3 \sqrt{xy} - x^3 y^2 \sqrt{xy} + x^6 \sqrt{xy}$

Factor out the common term $\sqrt{xy}$:

$\sqrt{xy} (x^2 y^3 - x^3 y^2 + x^6)$

Final Simplified Expression

$\boxed{\sqrt{xy} (x^2 y^3 - x^3 y^2 + x^6)}$

Would you like more details or have any questions?

Related Questions:

1. How do you simplify square roots involving variables?
2. What are the steps to factor expressions with common terms?
3. How can you simplify expressions with higher powers of variables?
4. How do you handle mixed radicals in simplification?
5. What are common mistakes to avoid when simplifying square roots?

Tip:

Always look for factors within the radicals that can be simplified separately to make the overall expression simpler!