To simplify $\sqrt{x^4 + x^3}$, let's see if any factoring can be applied inside the square root:

Step 1: Factor the expression inside the square root

We can factor out $x^3$ from both terms inside the square root: $\sqrt{x^4 + x^3} = \sqrt{x^3(x + 1)}$

Step 2: Apply the square root to the factors

We can now apply the square root to each factor individually: $\sqrt{x^3(x + 1)} = \sqrt{x^3} \cdot \sqrt{x + 1}$

Step 3: Simplify the square root of $x^3$

$\sqrt{x^3} = x^{3/2}$ Thus, the expression becomes: $x^{3/2} \cdot \sqrt{x + 1}$

So, the simplified form of $\sqrt{x^4 + x^3}$ is: $x^{3/2} \cdot \sqrt{x + 1}$

Would you like further clarification or details?

Here are 5 related questions to deepen your understanding:

1. How would you simplify $\sqrt{x^6 + x^3}$?
2. What is the difference between $\sqrt{x^3}$ and $(x^3)^{1/2}$?
3. Can the expression $\sqrt{x + 1}$ be simplified further?
4. How do square roots distribute over products but not sums?
5. How would you handle $\sqrt{x^4 - x^3}$?

Tip: When simplifying expressions inside a square root, always check for common factors to reduce complexity.