(i) (sqrt(a + x) + sqrt(a - x))/(sqrt(a + x) - sqrt(a - x)) = a/x

We are given the equation:

\[
\frac{\sqrt{a + x} + \sqrt{a - x}}{\sqrt{a + x} - \sqrt{a - x}} = \frac{a}{x}
\]

We will start by simplifying the left-hand side (LHS) using the difference of squares method.

### Step 1: Multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of \(\sqrt{a + x} - \sqrt{a - x}\) is \(\sqrt{a + x} + \sqrt{a - x}\). Therefore, multiply both the numerator and the denominator by \(\sqrt{a + x} + \sqrt{a - x}\):

\[
\frac{\left( \sqrt{a + x} + \sqrt{a - x} \right) \cdot \left( \sqrt{a + x} + \sqrt{a - x} \right)}{\left( \sqrt{a + x} - \sqrt{a - x} \right) \cdot \left( \sqrt{a + x} + \sqrt{a - x} \right)}
\]

### Step 2: Simplify the denominator using the difference of squares formula.
The denominator becomes:

\[
(\sqrt{a + x})^2 - (\sqrt{a - x})^2 = (a + x) - (a - x) = 2x
\]

### Step 3: Simplify the numerator.
The numerator is:

\[
\left( \sqrt{a + x} + \sqrt{a - x} \right)^2 = (\sqrt{a + x})^2 + 2\sqrt{(a + x)(a - x)} + (\sqrt{a - x})^2
\]
\[
= (a + x) + 2\sqrt{(a + x)(a - x)} + (a - x)
\]
\[
= a + x + a - x + 2\sqrt{(a + x)(a - x)}
\]
\[
= 2a + 2\sqrt{(a + x)(a - x)}
\]

### Step 4: Substitute the simplified numerator and denominator back into the expression.
We now have:

\[
\frac{2a + 2\sqrt{(a + x)(a - x)}}{2x}
\]

### Step 5: Cancel out the common factor of 2.
The expression simplifies to:

\[
\frac{a + \sqrt{(a + x)(a - x)}}{x}
\]

### Step 6: Simplify the square root term.
Using the difference of squares, we simplify \(\sqrt{(a + x)(a - x)}\):

\[
\sqrt{(a + x)(a - x)} = \sqrt{a^2 - x^2}
\]

### Step 7: Final expression.
The left-hand side becomes:

\[
\frac{a + \sqrt{a^2 - x^2}}{x}
\]

Since this is already in a simplified form, we compare it with the right-hand side, \(\frac{a}{x}\).

For the equation to hold true, the square root term must be 0, i.e., \(\sqrt{a^2 - x^2} = 0\).

Thus, \(a^2 - x^2 = 0\), which gives \(a^2 = x^2\), or \(a = x\) or \(a = -x\).

Hence, the equation holds when \(a = x\) or \(a = -x\).

---

Would you like further details or have any questions on the steps?

Here are some related questions you might explore:
1. What are the general properties of square roots in equations?
2. How can we manipulate expressions involving radicals?
3. Why is multiplying by a conjugate helpful in simplifying expressions?
4. What are other uses of the difference of squares technique?
5. How can we solve equations involving square roots in general?

**Tip:** Always check if the final solution satisfies the original equation by substituting back the values.

(sqrt(a + x) + sqrt(a - x))/(sqrt(a + x) - sqrt(a - x)) = a/x

Learn how to simplify the expression (sqrt(a + x) + sqrt(a - x))/(sqrt(a + x) - sqrt(a - x)) using the difference of squares method. This solution involves algebraic techniques, including simplifying radicals and applying the difference of squares formula. Suitable for high school students in Grades 10-12.

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