**Simplify radical expressions by rationalizing the denominator :**

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.

**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).

**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.

**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 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**

**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

**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.

**Example 5 :**

**Rationalize the denominator **

**Solution :**

**To rationalize the denominator, we have to multiply **√(30 x) by the numerator and denominator.

When we have two same numbers or variables inside the radical, we can take one number and variable out of the radical sign.

- Rationalizing the denominator worksheet
- Logarithms
- Permutation and combination problems
- LCM and HCF word problems
- Time and distance word problems
- Mensuration
- Trigonometry
- Integration

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