To verify whether the equation given is true, we need to check if both sides simplify to the same value. The equation is:

$\frac{5}{\sqrt{3} - 7} = \frac{\sqrt{3} + 7}{\sqrt{3} + 7}$

First, let's simplify the right-hand side:

Since the numerator and denominator are the same, $\frac{\sqrt{3} + 7}{\sqrt{3} + 7} = 1$.

Now, we simplify the left-hand side. We'll multiply the numerator and the denominator by the conjugate of the denominator to eliminate the square root:

$\frac{5}{\sqrt{3} - 7} \cdot \frac{\sqrt{3} + 7}{\sqrt{3} + 7} = \frac{5(\sqrt{3} + 7)}{(\sqrt{3} - 7)(\sqrt{3} + 7)}$

Calculating the denominator using the difference of squares:

$(\sqrt{3} - 7)(\sqrt{3} + 7) = (\sqrt{3})^2 - 7^2 = 3 - 49 = -46$

The equation becomes:

$\frac{5(\sqrt{3} + 7)}{-46}$

Expanding the numerator:

$\frac{5\sqrt{3} + 35}{-46}$

This fraction is not equal to 1, so the original equation:

$\frac{5}{\sqrt{3} - 7} = \frac{\sqrt{3} + 7}{\sqrt{3} + 7}$

is not true. The correct value of the left-hand side is $\frac{5\sqrt{3} + 35}{-46}$.

Would you like more details on this solution, or do you have any other questions?

Related Questions:

1. How do you simplify complex rational expressions?
2. What is the importance of using the conjugate in rational expressions involving square roots?
3. Can you explain the difference of squares formula and its applications?
4. How do rational expressions relate to real-life situations?
5. What are other methods to check if two expressions are equivalent?

Tip: When working with square roots in fractions, multiplying by the conjugate often simplifies the expression and helps remove radicals from the denominator.