**How to find simplest rationalizing factor ? :**

Here we are going to how to find the simplest rationalizing factor.

If the product of two irrational numbers is rational, then each one is called the rationalizing factor of the other

- (a + √x) and (a + √x) are rationalizing factors of each other.
- (a + b√x) and (a - b√x) are rationalizing factors of each other.
- √x + √y and √x - √y are rationalizing factors of each other.

**Example 1 :**

Write the rationalizing factor of the following

3√2

**Solution :**

**3√2 is irrational number.**

**By multiplying **3√2 by √2, we get rational number.

= 3√2 ⋅ √2

= 3√(2 ⋅ 2)

Since we have two same numbers multiplying inside the radical, we can factor out one term.

= 3 ⋅ 2

= 6 is the rational number

Hence the rationalizing factor of 3√2 is √2.

**Example 2 :**

Write the rationalizing factor of the following

2 ∛5

**Solution :**

2 ∛5** is irrational number.**

**By multiplying **2 ∛5 by ∛25, we may get rid of the cube root.

= 2 ∛5 ⋅ ∛25

= 2 ∛(5 ⋅ 25)

= 2 ∛(5 ⋅ 5 ⋅ 5)

= 2 ⋅ 5

= 10 is rational number

Hence the rationalizing factor of 2 ∛5 is ∛25.

**Example 3 :**

Write the rationalizing factor of the following

5 - 4√3

**Solution :**

Rationalizing factor of 5 - 4√3 is 5 + 4√3

(5 - 4√3) (5 + 4√3) = 5(5) + 5(4√3) - 4√3(5) - 16(√3√3)

= 25 + 20 √3 - 20 √3 - 16 (3)

= 25 - 48

= -23 it is rational number

Hence the rationalizing factor of 5 - 4√3 is 5 + 4√3

**Example 4 :**

Write the rationalizing factor of the following

√2 + √3

**Solution :**

Rationalizing factor of √2 + √3 is √2 - √3

**Example 5 :**

Write the rationalizing factor of the following

√5 - √2

**Solution :**

Rationalizing factor of √5 - √2 is √5 + √2

**Example 6 :**

Write the rationalizing factor of the following

2 + √3

**Solution :**

Rationalizing factor of 2 + √3 is 2 - √3

Let us look into the next example on "How to find simplest rationalizing factor".

**Example 7 :**

Write the rationalizing factor of the following

√75

**Solution :**

To find the least number to be multiplied to convert √75 as rational number.We have to split 75.

√75 = √(5 ⋅ 5 ⋅ 3)

√75 ⋅ √3 = √(5 ⋅ 5 ⋅ 3) ⋅ √3

= √(5 ⋅ 5 ⋅ 3 ⋅ 3)

= 15 is the rational number

Hence √3 is the rationalizing factor of √75.

Let us look into the next example on "How to find simplest rationalizing factor".

**Example 8 :**

Write the rationalizing factor of the following

5 - 7√3

**Solution :**

Rationalizing factor of 5 - 7√3 is 5 + 7√3

- Rationalization of surds
- Comparison of surds
- Operations with radicals
- Ascending and descending order of surds
- Simplifying radical expression
- Exponents and powers

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