Real Numbers Algebra: Laws of Square Roots, Rationalizing Denominators, and Conjugate Pairs
Welcome to Unit 2 of Mathematics for Preparatory 2! In this foundational lesson, we master algebraic operations on the set of Real Numbers R, focusing on 'Square Roots' and 'Conjugate Numbers'. For any non-negative real numbers a and b, the fundamental product law states that sqrt(a) * sqrt(b) = sqrt(a * b), and the quotient law states that sqrt(a) / sqrt(b) = sqrt(a / b) (where b > 0). We learn to simplify radicals into their simplest form by factoring out perfect squares (e. g., sqrt(50) = sqrt(25 * 2) = 5*sqrt(2)). Furthermore, we explore 'Conjugate Pairs' (sqrt(a) + sqrt(b)) and (sqrt(a) - sqrt(b)), discovering that their product is always a purely rational number: (sqrt(a))^2 - (sqrt(b))^2 = a - b. This powerful identity enables us to rationalize binomial denominators effortlessly.
1. Laws of Square Roots
Fundamental algebraic properties for simplifying radicals:
- Multiplication Law: √a × √b = √(a × b), where a ≥ 0 and b ≥ 0.
- Division Law: √a ÷ √b = √(a / b), where a ≥ 0 and b > 0.
- Extracting Perfect Squares: √(a² × b) = |a|√b.
• √18 = √(9 × 2) = 3√2.
• √75 = √(25 × 3) = 5√3. - Combining Like Radicals: 4√3 + 5√3 = 9√3, whereas √2 + √3 cannot be combined into √5.
2. Conjugate Numbers & Rationalizing
Eliminating irrational radicals from fraction denominators:
- Monomial Denominator: Multiply numerator and denominator by the radical:
• 6 / √3 = (6 × √3) / (√3 × √3) = 6√3 / 3 = 2√3. - Conjugate Pairs Definition: The two expressions (√a + √b) and (√a - √b) are conjugates.
• Product: (√a + √b)(√a - √b) = (√a)² - (√b)² = a - b ∈ Q.
• Sum: (√a + √b) + (√a - √b) = 2√a. - Binomial Rationalizing: Multiply both terms by the conjugate of the denominator to eliminate the radical.
Step-by-Step Problem Solving Workshop:
Solution:
1. √32 = √(16 × 2) = 4√2.
2. √50 = √(25 × 2) = 5√2.
3. Expression = 4√2 + 5√2 - 3√2 = (4 + 5 - 3)√2 = 6√2.
Solution:
1. x = [4(√7 + √3)] / [(√7 - √3)(√7 + √3)] = [4(√7 + √3)] / (7 - 3) = [4(√7 + √3)] / 4 = √7 + √3.
2. Since y = √7 - √3, x and y are conjugates.
3. x + y = (√7 + √3) + (√7 - √3) = 2√7.
4. (x + y)² = (2√7)² = 4 × 7 = 28.
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