Conjugate Surds

The sum and difference of two simple quadratic surds are said to be conjugate surds to each other.

Conjugate surds are also known as complementary surds.

Thus, the sum and the difference of two simple quadratic surds 4√7and √2 are 4√7 + √2 and 4√7 - √2   respectively. Therefore, two surds (4√7 + √2) and (4√7 - √2) are conjugate to each other.

Similarly, two surds (-2√5 + √3) and (-2√5 - √3) are conjugate to each other.

In general, two binomial quadratic surds (x√a + y√b) and (x√a - y√b) are conjugate to each other.

Note:

1. Since 3 + √5 = √9 + √5 and surd conjugate to √9 + √5 is √9 - √5, hence it is evident that surds 3 + √5 and 3 - √5 are conjugate to each other.

In general, surds (a + x√b) and (a - x√b) are complementary to each other.

2. The product of two binomial quadratic surds is always rational.

For example,

(√m + √n)(√m - √n) = (√m)^2 - (√n)^2 = m - n, which is rational.

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