When the denominator of an expression contains a term with a square root or a number under radical sign, the process of converting into an equivalent expression whose denominator is a rational number is called rationalizing the denominator.
If the product of two irrational numbers is rational, then each one is called the rationalizing factor of the other.
Case 1 :
If the denominator is in the form of √a (where a is a rational number).
Then we have to multiply both the numerator and denominator by the same (√a).
Example 1 :
Rationalize the denominator 18/√6
Solution :
Step 1 :
We have to rationalize the denominator. Here we have √6 (in the form of √a). Then we have to multiply the numerator and denominator by √6
Step 2 :
By multiplying the numerators and denominators of first and second fraction , we get
Step 3 :
By simplifications, we get 3√6
Case 2 :
If the denominator is in the form of a ± √b or a ± c √b (where b is a rational number).
Then we have to multiply both the numerator and denominator by its conjugate.
a + √b and a - √b are conjugate of each other.
a + c√b and a - c√b are conjugate of each other.
Example 2 :
Rationalize the denominator 4/(1+2√3)
Solution :
Step 1 :
Here we have (1+2√3) (in the form of a + c√b) in the denominator. Then we have to multiply the numerator and denominator by the conjugate of (1+2√3).
Conjugate of (1 + 2√3) is (1 - 2√3)
Step 2 :
By multiplying the numerators and denominators of first and second fraction, we get
Step 3 :
By comparing the denominator with the algebraic identity a2 - b2 = (a + b)(a - b), we get
Case 3 :
If the denominator is in the form of √a ± √b (where a and b are rational numbers).
Then we have to multiply both the numerator and denominator by its conjugate.
√a + √b and √a - √b are conjugate of each other.
Example 3 :
Rationalize the denominator (6 + √5)/(6-√5)
Solution :
Step 1 :
Here we have (6-√5) in the denominator. Then we have to multiply the numerator and denominator by the conjugate of (6-√5).
Conjugate of (6-√5) is (6+√5)
Example 4 :
Rationalize the denominator (2 + √3)/(2 - √3) = x + y√3 and find the value of x and y.
Solution :
Now we have to compare the final answer with R.H.S
The values of x and y are 7 and 4 respectively.
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