To simplify the expression $(\sqrt{5} - \sqrt{7})^2$, we will use the formula for the square of a binomial:

$(a - b)^2 = a^2 - 2ab + b^2$

Here, $a = \sqrt{5}$ and $b = \sqrt{7}$. Substituting these values into the formula, we get:

$(\sqrt{5} - \sqrt{7})^2 = (\sqrt{5})^2 - 2(\sqrt{5})(\sqrt{7}) + (\sqrt{7})^2$

Now, let's compute each term:

1. $(\sqrt{5})^2 = 5$
2. $(\sqrt{7})^2 = 7$
3. $2 \cdot \sqrt{5} \cdot \sqrt{7} = 2\sqrt{35}$

Putting these together, we have:

$(\sqrt{5} - \sqrt{7})^2 = 5 - 2\sqrt{35} + 7$

Simplify by combining like terms:

$5 + 7 - 2\sqrt{35} = 12 - 2\sqrt{35}$

So, the simplified form of $(\sqrt{5} - \sqrt{7})^2$ is:

$\boxed{12 - 2\sqrt{35}}$

Would you like more details or have any questions?

Related Questions:

1. How do you find the square of a binomial with mixed square roots?
2. What is the difference between expanding $(a + b)^2$ and $(a - b)^2$?
3. Can you explain how to simplify expressions involving square roots further?
4. How do you multiply and simplify expressions with different radicals?
5. What are some applications of squaring binomials in algebra?

Tip: When squaring a binomial with radicals, always apply the formula carefully to avoid mistakes with the middle term.