This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
The whiteboard discusses the topic of Surds, specifically focusing on Division or Rationalization of Surds. Here's an explanation of the concepts and solutions to the examples shown: 1. What is a Surd? A surd is an irrational number that can be expressed as the root of an integer, where the root cannot be simplified to a whole number. For example, sqrt(2), sqrt(3), sqrt(5) are surds, but sqrt(4) is not a surd because sqrt(4) = 2. 2. Rationalization of Surds Rationalization is the process of converting a fraction with an irrational number (a surd) in its denominator into an equivalent fraction with a rational number in its denominator. This is done to simplify expressions and make calculations easier, as it's generally preferred not to have surds in the denominator. 3. How to Rationalize a Denominator with a Single Surd If the denominator is a single surd, like sqrt(a), you multiply both the numerator and the denominator by that surd. This is because sqrt(a) × sqrt(a) = a, which is a rational number. 4. Conjugate of a Surd The conjugate of a surd expression is used when the denominator is a binomial (two terms) involving a surd, such as a + sqrt(b) or sqrt(a) + sqrt(b). The conjugate is formed by changing the sign between the terms. The conjugate of a + sqrt(b) is a - sqrt(b). The conjugate of a - sqrt(b) is a + sqrt(b). When an expression is multiplied by its conjugate, the result is a rational number, because (x+y)(x-y) = x^2 - y^2. For example, (a + sqrt(b))(a - sqrt(b)) = a^2 - (sqrt(b))^2 = a^2 - b. --- Examples from the Whiteboard: Here's how to rationalize the denominators for the given examples: a) (1)/(sqrt(2)) To rationalize, multiply the numerator and denominator by sqrt(2): (1)/(sqrt(2)) × sqrt(2)sqrt(2) = 1 × sqrt(2)sqrt(2) × sqrt(2) = sqrt(2)2 The rationalized form is sqrt(2)2. b) (1)/(sqrt(3)) To rationalize, multiply the numerator and denominator by sqrt(3): (1)/(sqrt(3)) × sqrt(3)sqrt(3) = 1 × sqrt(3)sqrt(3) × sqrt(3) = sqrt(3)3 The rationalized form is sqrt(3)3. c) (1)/(sqrt(5)) To rationalize, multiply the numerator and denominator by sqrt(5): (1)/(sqrt(5)) × sqrt(5)sqrt(5) = 1 × sqrt(5)sqrt(5) × sqrt(5) = sqrt(5)5 The rationalized form is sqrt(5)5. d) (1)/(sqrt(7)) To rationalize, multiply the numerator and denominator by sqrt(7): (1)/(sqrt(7)) × sqrt(7)sqrt(7) = 1 × sqrt(7)sqrt(7) × sqrt(7) = sqrt(7)7 The rationalized form is sqrt(7)7. e) (1)/(sqrt(11)) To rationalize, multiply the numerator and denominator by sqrt(11): (1)/(sqrt(11)) × sqrt(11)sqrt(11) = 1 × sqrt(11)sqrt(11) × sqrt(11) = sqrt(11)11 The rationalized form is sqrt(11)11.