Express the product left parenthesis, square root of, 5, end square root, plus, 7, right parenthesis, left parenthesis, square root of, 5, end square root, minus, 7, right parenthesis(5​+7)(5​−7) in simplest form.

The given expression is:

\[
(\sqrt{5} + 7)(\sqrt{5} - 7)
\]

This is in the form of the difference of squares:

\[
(a + b)(a - b) = a^2 - b^2
\]

Here, \(a = \sqrt{5}\) and \(b = 7\). Applying the difference of squares formula:

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

Now, simplify:

\[
(\sqrt{5})^2 = 5
\]
\[
7^2 = 49
\]

Thus:

\[
5 - 49 = -44
\]

So, the simplest form of the expression is:

\[
\boxed{-44}
\]

Would you like further clarification on this? Here are five related questions to consider:

1. What is the difference of squares formula used in algebra?
2. How would this expression change if the terms were \((\sqrt{5} + 7)(\sqrt{5} + 7)\)?
3. How do you simplify square roots in general?
4. What are some other algebraic identities similar to the difference of squares?
5. How can this concept be applied to larger numbers or variables?

**Tip:** Always check if an expression fits into a known algebraic identity, such as difference of squares, to simplify it quickly.

Express the product (√5 + 7)(√5 - 7) in simplest form.

Explore how to simplify the product (√5 + 7)(√5 - 7) using the difference of squares formula. This detailed step-by-step solution explains the algebraic concepts of simplifying radicals and applying a key algebraic identity, suitable for students in Grades 8-10.

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