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.

Learn how to simplify sqrt(x^4 + x^3) with step-by-step factoring and application of square root properties. This problem covers key algebraic concepts like radicals and exponent rules. Suitable for high school students in Grades 9-12.

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