find the rationalising factor of 1/3√5 and 1/√12
Answers
When the product of two surds is a rational number, then each of the two surds is called rationalizing factor of the other.
For example, 5√2√ is a surd where 5–√ is numerator and 2–√ is denominator. Now for rationalization of surds, if we multiply both numerator and denominator by 2–√, then denominator will be a rational number.
5√2√ = 5√×2√2√×2√
= 10√2.
So after rationalization of surd 5√2√, it is becoming 10√2 where 2–√ is used as rationalization factor.
In other words, if the product of two surds is rational, then each is called a rationalizing factor of the other and each is said to be rationalized by the other.
For complex surds where the surds are in order 2, conjugates are used to rationalize the surds. This comes the from the formula a2−b2=(a+b)(a−b). So in complex surds order 2 surds get squared off and denominators are converted to a rational number.
Like for example, if rationalization of the complex surd 12√2−1 to be done, denominator 2–√2−1 is to converted to a rational number. If a = 2–√2 and b = 1, then denominator is (a - b), if we multiply (a + b), it will a2−b2 and 2–√2 will be squared off.
12√2−1
= (2√2+1)(2√2−1)(2√2+1)
= (2√2+1)2−1
= 2–√2 + 1.
For complex surds in the denominator in other forms or in order more than 2, can be rationalized by using suitable multiplication factors.
Examples of rationalization of surds:
1. For example, the rationalizing factor of √5 is √5 and rationalizing factor of ∛2 is ∛2^2 or ∛4. Since, √5 × √5 = 5 and ∛2 × ∛2^2 = ∛(2 × 2^2) = ∛2^3 = 2
2. (a√z) × (b√z) = (a × b) × (√z × √z) = ab(√z)^2 = abz, which is rational. Therefore, each of the surds a√z and b√z is a rationalizing factor of the other.
3. √5 × 2√5 = 2 × (√5)^2 = 3 × 5 = 15, which is rational. Therefore, each of the surds √5 and 2√5 is a rationalizing factor of the other.
4. (√a + √b) × (√a - √b) = (√a)^2 - (√b)^2 = a - b, which is rational. Therefore, each of the surds (√a + √b) and (√a - √b) is a rationalizing factor of the other.
5. (x√a + y√b) × (x√a - y√b) = (x√a)^2 - (y√b)^2 = ax - by, which is rational. Therefore, each of the surds (x√a + y√b) and (x√a - y√b) is a rationalizing factor of the other.
6. (4√7 + √3) × (4√7 - √3) = (4√7)^2 - (√3)^2 = 112 - 3 = 109, which is rational. Therefore, each of the surd factors (4√7 + √3) and (4√7 - √3) is a rationalizing factor of the other.
7. Also rationalizing factor of ∛(ab^2c^2) is ∛(a^2bc) because ∛(ab^2c^2) × ∛(a^2bc) = abc.
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