**Simplifying radical expressions with conjugates worksheet :**

Whenever we want to simplify radical expressions using conjugates, first we have to consider the denominator. If the denominators are in one of following forms, then we have to do the following steps for simplification.

Here we are going to see some practice questions of this topic.

(1) Rationalize the denominator 1/(2 + √5)

(2) Rationalize the denominator (6 + √5)/(6 - √5)

(3) Rationalize the denominator 1/(8 - 2 √5)

**Question 1 :**

Rationalize the denominator 1/(2 + √5)

**Solution :**

**Because we have the denominator 2 + **√5, we have to multiply by its conjugate **2 - **√5.

**By multiplying the numerators, we get 2 - **√5 in denominator we get (**2 + **√5) (**2 - **√5).

Comparing the denominator with the algebraic identity (a** + b**) (a** - b**), we have "2" instead of "a" and "√5" instead of b.

(a** + b**) (a** - b**) = a² - b²

That is,

(**2 + **√5) (**2 - **√5) = 2² - √5²

= 4 - 5

= -1

"Simplifying radical expressions with conjugates worksheet

By distributing negative sign to the numerator, we get -2 + √5

Hence the answer is -2 + √5.

**Question 2 :**

Rationalize the denominator (6 + √5)/(6 - √5)

**Solution :**

**Step 1 :**

**Because we have the denominator **(6 - √5), we have to multiply by its conjugate (6 + √5)

= [(6 + √5)/(6 - √5)] x [(6 + √5)/(6 + √5)]

**Step 2 :**

**By multiplying the numerators, we get **(6 + √5) (6 + √5), in denominator we get (6 - √5) (6 + √5)

= [(6 + √5)²/(6 - √5)(6 + √5)]

**Step 3 :**

Comparing the denominator with the algebraic identity (a** + b**)², we have "6" instead of "a" and "√5" instead of b.

(a** + b**)² = a² + 2ab + b²

(6 + √5)² = 6² + 2(6) (√5) + √5²

= 36 + 12√5 + 5

= 41 + 12√5

Comparing the denominator with the algebraic identity (a** + b**) (a** - b**), we have "6" instead of "a" and "√5" instead of b.

(a** + b**) (a** - b**) = a² - b²

That is,

(6 - √5)(6 + √5) = 6² - √5²

= 36 - 5

= 31

**Step 4 :**

Hence the answer is (41 + 12√5) / 31

**Question 3 :**

Rationalize the denominator 1/(8 - 2 √5)

**Solution :**

**Step 1 :**

**Because we have the denominator **(8 - 2 √5), we have to multiply by its conjugate (8 + 2 √5)

= [1/(8 - 2 √5)] x [(8 + 2 √5)/(8 + 2 √5)]

**Step 2 :**

**By multiplying the numerators, we get **(8 + 2 √5), in denominator we get (8 + 2 √5) (8 - 2 √5)

**Step 3 :**

Comparing the denominator with the algebraic identity (a** + b**) (a** - b**), we have "8" instead of "a" and "2√5" instead of b.

(a** + b**) (a** - b**) = a² - b²

That is,

(8 + 2 √5) (8 - 2 √5) = 8² - (2√5)²

= 64 - 2²√5²

= 64 - 4(5)

= 64 - 20

= 44

**Step 4 :**

= (8 + 2 √5) / 44

By factoring 2, we get 2(4 - √5) / 44

= (4 + √5) / 22

Hence the answer is (4 + √5) / 22

After having gone through the stuff given above, we hope that the students would have understood "Simplifying radical expressions with conjugates worksheet".

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