To prove that the equation 

\[
\sqrt{n+3} - \sqrt{n} = \frac{3}{\sqrt{n+3} + \sqrt{n}}
\]

holds true for all integers \( n \geq 1 \), follow these steps:

### Step 1: Rationalize the Left-Hand Side
Start by focusing on the left-hand side (LHS) of the equation:

\[
\sqrt{n+3} - \sqrt{n}
\]

To simplify this expression, multiply the numerator and the denominator by the conjugate of the expression:

\[
\sqrt{n+3} - \sqrt{n} \times \frac{\sqrt{n+3} + \sqrt{n}}{\sqrt{n+3} + \sqrt{n}}
\]

This gives:

\[
\frac{(\sqrt{n+3} - \sqrt{n})(\sqrt{n+3} + \sqrt{n})}{\sqrt{n+3} + \sqrt{n}}
\]

### Step 2: Simplify the Numerator
Apply the difference of squares formula to the numerator:

\[
(\sqrt{n+3})^2 - (\sqrt{n})^2 = (n+3) - n = 3
\]

So, the expression simplifies to:

\[
\frac{3}{\sqrt{n+3} + \sqrt{n}}
\]

### Step 3: Compare with the Right-Hand Side
Notice that the simplified LHS is exactly the same as the right-hand side (RHS) of the original equation:

\[
\frac{3}{\sqrt{n+3} + \sqrt{n}}
\]

Since both sides are equal, the equation is proven to be true for all \( n \geq 1 \).

Would you like further details on this proof or any additional questions?

### Related Questions:
1. What is the difference of squares, and why is it useful in simplifying square roots?
2. Can you think of other examples where rationalizing the denominator is beneficial?
3. How does multiplying by the conjugate affect the expression?
4. What are some potential pitfalls when working with square roots in algebraic expressions?
5. Could you apply this method to any expression of the form \( \sqrt{a+b} - \sqrt{a} \)?

### Tip:
Remember that when simplifying expressions involving roots, using identities such as the difference of squares can greatly simplify your work.

Explore how to prove the equation √(n+3) - √(n) = 3 / (√(n+3) + √(n)) for all integers n ≥ 1 with a detailed step-by-step solution. Learn about rationalizing denominators and applying the difference of squares in algebraic expressions. Suitable for advanced high school students and anyone interested in algebraic proofs.

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