**Conjugate of Surds**

**Definition:**
The conjugate of a surd is obtained by changing the sign (+ to - or - to +) of the irrational part of the surd. In other words, if a surd is of the form a + √b, then its conjugate is a - √b.

**Example:**
Let's take the surd √5 + 2. The conjugate of this surd will be √5 - 2.

**Properties of Conjugates:**
1. The product of a surd and its conjugate is always a rational number.
2. The sum or difference of a surd and its conjugate is always a rational number.

**Proof of Properties:**
1. Let's consider a surd a + √b and its conjugate a - √b. The product of these two surds is (a + √b)(a - √b) = a^2 - b. Here, a^2 is a rational number and -b is also a rational number. Therefore, the product of a surd and its conjugate is always a rational number.
2. Now, let's consider a surd a + √b and its conjugate a - √b. The sum of these two surds is (a + √b) + (a - √b) = 2a. Here, 2a is a rational number. Similarly, the difference of these two surds is (a + √b) - (a - √b) = 2√b. Here, 2√b is also a rational number. Therefore, the sum or difference of a surd and its conjugate is always a rational number.

**Application:**
The concept of conjugates of surds is useful in simplifying and rationalizing expressions involving surds.

**Example:**
Let's consider the expression (√3 + √2)(√3 - √2). Using the formula (a + b)(a - b) = a^2 - b^2, we can simplify the expression as (√3)^2 - (√2)^2 = 3 - 2 = 1, which is a rational number.

**Conclusion:**
The conjugate of a surd is obtained by changing the sign of its irrational part. The product of a surd and its conjugate is always a rational number, and the sum or difference of a surd and its conjugate is also a rational number. The concept of conjugates of surds is helpful in simplifying and rationalizing expressions involving surds.