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 product of conjugates, which follows the pattern \((a - b)(a + b) = a^2 - b^2\).
2. Identify \(a = \sqrt{2}\) and \(b = \sqrt{3}\) from the given expression.
3. Apply the difference of squares formula: \(a^2 - b^2 = (\sqrt{2})^2 - (\sqrt{3})^2\).
4. Calculate each square separately: \((\sqrt{2})^2 = 2\) and \((\sqrt{3})^2 = 3\).
5. Subtract the results to get the simplified expression: \(2 - 3\).

Key Concepts

Difference of Squares

The difference of squares is a special product formula: (a - b)(a + b) = a² - b². It simplifies multiplication by converting the product of conjugates into the difference between the squares of the two terms.

Simplifying Square Roots

Simplifying square roots involves expressing radicals in their simplest form, often by factoring out perfect squares. This helps in performing arithmetic operations and recognizing patterns like the difference of squares.

Multiplying Binomials

Multiplying binomials requires applying the distributive property (FOIL method) to combine each term in the first binomial with each term in the second. This process expands the product into a polynomial expression.