Multiply. See Example 7.(√2 - √3) (√2 + √3)

Verified step by step guidance

1. Recognize that the expression \((\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3})\) is in the form of \((a - b)(a + b)\), which is a difference of squares.
2. Recall the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\).
3. Identify \(a = \sqrt{2}\) and \(b = \sqrt{3}\) in the expression.
4. Apply the difference of squares formula: \((\sqrt{2})^2 - (\sqrt{3})^2\).
5. Calculate each square: \((\sqrt{2})^2 = 2\) and \((\sqrt{3})^2 = 3\), then subtract the results.

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity that states that the product of two conjugates, (a - b)(a + b), equals a² - b². In the given expression, (√2 - √3)(√2 + √3), we can apply this identity to simplify the multiplication, resulting in 2 - 3.

Radicals

Radicals are expressions that involve roots, such as square roots. In this problem, √2 and √3 are both radicals. Understanding how to manipulate and simplify expressions involving radicals is essential for solving problems that include them, especially when combined with other algebraic operations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms and eliminating unnecessary components. In this case, after applying the difference of squares, we simplify the result to -1, demonstrating the importance of simplification in algebraic operations.