Perform the operation and/or simplify each of the following. Assume all variables represent positive real numbers. (2 + √3) (2 - √3)

검증된 단계별 안내

Recognize that the expression \((2 + \sqrt{3})(2 - \sqrt{3})\) is a product of conjugates, which follows the pattern \((a + b)(a - b) = a^2 - b^2\).

Identify \(a = 2\) and \(b = \sqrt{3}\) in the expression.

Apply the difference of squares formula: \(a^2 - b^2 = 2^2 - (\sqrt{3})^2\).

Calculate each square separately: \(2^2 = 4\) and \((\sqrt{3})^2 = 3\).

Subtract the squares to simplify the expression: \(4 - 3\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Difference of Squares

The difference of squares is a special product formula: (a + b)(a - b) = a² - b². It simplifies multiplication of conjugates by subtracting the square of the second term from the square of the first, eliminating the middle terms.

Simplifying Square Roots

Simplifying square roots involves expressing radicals in their simplest form by factoring out perfect squares. This helps in combining like terms and performing arithmetic operations more easily.

Properties of Real Numbers

Understanding that variables represent positive real numbers ensures that operations like square roots are defined and results are real. This context allows simplification without considering complex numbers.