The given expression is:

\[
\frac{\sqrt{59.29} - \sqrt{5.29}}{\sqrt{59.29} + \sqrt{5.29}}
\]

To simplify it, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{59.29} - \sqrt{5.29} \). This eliminates the square roots in the denominator.

\[
\frac{(\sqrt{59.29} - \sqrt{5.29})(\sqrt{59.29} - \sqrt{5.29})}{(\sqrt{59.29} + \sqrt{5.29})(\sqrt{59.29} - \sqrt{5.29})}
\]

### Step-by-step simplification:

1. **Denominator:**

\[
(\sqrt{59.29} + \sqrt{5.29})(\sqrt{59.29} - \sqrt{5.29}) = (\sqrt{59.29})^2 - (\sqrt{5.29})^2
\]
\[
= 59.29 - 5.29 = 54
\]

2. **Numerator:**

\[
(\sqrt{59.29} - \sqrt{5.29})^2 = (\sqrt{59.29})^2 - 2\sqrt{59.29} \cdot \sqrt{5.29} + (\sqrt{5.29})^2
\]
\[
= 59.29 - 2\sqrt{59.29 \cdot 5.29} + 5.29
\]
\[
= 59.29 + 5.29 - 2\sqrt{313.5841}
\]
\[
= 64.58 - 2 \times 17.71 = 64.58 - 35.42 = 29.16
\]

Thus, the simplified expression is:

\[
\frac{29.16}{54}
\]

Which gives approximately:

\[
\frac{29.16}{54} \approx 0.54
\]

So, the simplified value of the expression is approximately \( 0.54 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How does multiplying by the conjugate simplify expressions with square roots?
2. Can this technique be applied to any expression involving square roots?
3. What is the significance of rationalizing the denominator?
4. How can you simplify similar expressions with irrational numbers?
5. How is this approach connected to difference of squares?

**Tip:** Always consider using the conjugate for expressions involving square roots in the denominator to simplify calculations.

Simplify the expression √59.29 - √5.29 / √59.29 + √5.29 by multiplying by the conjugate. This detailed solution explores algebraic concepts such as radicals, conjugates, and the difference of squares. Suitable for high school students (Grades 9-12).

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