You want to simplify the following expression:

$\frac{x\sqrt{x-1}}{x-1} - \frac{x+3}{\sqrt{x}+1}$

Let's proceed step by step:

Step 1: Analyze the first term $\frac{x\sqrt{x-1}}{x-1}$

The first term has a square root, and it can be simplified by recognizing that the square root could simplify with the denominator if we manipulate it. We can write:

$\frac{x\sqrt{x-1}}{x-1} = \frac{\sqrt{x-1} \cdot x}{x-1}$

It is worth noting that there is no immediate simplification without further manipulation, so let’s move to the next term and then attempt to simplify both at once.

Step 2: Analyze the second term $\frac{x+3}{\sqrt{x}+1}$

The second term involves a rational expression with a square root in the denominator. To rationalize this, we multiply both the numerator and denominator by $\sqrt{x}-1$ (the conjugate of $\sqrt{x}+1$):

$\frac{x+3}{\sqrt{x}+1} \cdot \frac{\sqrt{x}-1}{\sqrt{x}-1} = \frac{(x+3)(\sqrt{x}-1)}{(\sqrt{x}+1)(\sqrt{x}-1)}$

Using the difference of squares formula in the denominator:

$(\sqrt{x}+1)(\sqrt{x}-1) = x - 1$

So, the second term becomes:

$\frac{(x+3)(\sqrt{x}-1)}{x-1}$

Step 3: Combine both terms

Now, we combine the two simplified terms:

$\frac{\sqrt{x-1} \cdot x}{x-1} - \frac{(x+3)(\sqrt{x}-1)}{x-1}$

Since both terms have the same denominator, we can combine the numerators:

$\frac{x\sqrt{x-1} - (x+3)(\sqrt{x}-1)}{x-1}$

Step 4: Expand and simplify the numerator

Expand $(x+3)(\sqrt{x}-1)$:

$(x+3)(\sqrt{x}-1) = (x+3)\sqrt{x} - (x+3)$

Now substitute this back into the numerator:

$x\sqrt{x-1} - (x+3)\sqrt{x} + (x+3)$

Final Expression

Thus, the simplified form of the original expression is:

$\frac{x\sqrt{x-1} - (x+3)\sqrt{x} + (x+3)}{x-1}$

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Would you like more detailed steps on any part of the process or further simplifications? Let me know if you need clarification!

Here are some related questions:

1. How can you simplify square roots in rational expressions?
2. How do you rationalize denominators with square roots?
3. Can you solve for $x$ if this expression equals zero?
4. What happens if you substitute specific values for $x$ (like $x=2$) into the expression?
5. How do you handle complex expressions with multiple square roots in algebra?

Tip: When simplifying expressions with square roots, try to look for opportunities to rationalize or combine terms by factoring or using conjugates to eliminate square roots from the denominator.