We will organize calculations with square roots and learn rationalization of denominators.
Basics of Square Roots
- (√a)² = a (a ≧ 0)
- √a × √b = √(ab)
- √a / √b = √(a/b)
- √(a²) = |a|
Simplifying Radicals
Take perfect-square factors outside the radical. √12 = 2√3, √50 = 5√2.
Example ①
(1) √48 (2) √27 + √12 (3) √8 × √18
Answer:
(1) 4√3 (2) 3√3 + 2√3 = 5√3 (3) √144 = 12
(1) √48 (2) √27 + √12 (3) √8 × √18
Answer:
(1) 4√3 (2) 3√3 + 2√3 = 5√3 (3) √144 = 12
Rationalization of the Denominator
For 1/√a, multiply the numerator and denominator by √a. For 1/(√a + √b), multiply the numerator and denominator by √a − √b.
Example ②
(1) 3/√2 (2) 1/(√5 − √3)
Answer: (1) 3√2/2 (2) (√5 + √3)/2
(1) 3/√2 (2) 1/(√5 − √3)
Answer: (1) 3√2/2 (2) (√5 + √3)/2
Key Point
- √(a²) = |a|
- Take perfect-square factors outside the radical
- When rationalizing a denominator, choose the expression to multiply by according to the denominator
Practice Problems
- Calculate √75 − √27 + √48.
- Rationalize the denominator of 2/(√3 + 1).
- Expand (√2 + √3)².
Answers and Explanations
- Answer: 6√3
Explanation: 5√3 − 3√3 + 4√3 = 6√3. - Answer: √3 − 1
Explanation: Multiply the numerator and denominator by √3 − 1: 2(√3−1)/2 = √3 − 1. - Answer: 5 + 2√6
Explanation: 2 + 2√6 + 3 = 5 + 2√6.