SIMPLIFYING RADICAL EXPRESSIONS : MIXED REVIEW

How to simplify radical terms ?

Example 1 : 

√a ⋅ √a  =  a

Example 2 : 

√(a ⋅ a)  =  a

Example 3 : 

√a ⋅ √b  =  √(ab)

Example 4 :

√a / √b  =  √(a/b)

Example 5 : 

√a + √a  =  2√a

Example 6 : 

3√a - 2√a  =  √a

Mixed Review

Question 1 :

Simplify : 

√5 ⋅ √18

Solution :

√5 ⋅ √18  =  √5 ⋅ √(3 ⋅ 3 ⋅ 2)

√5 ⋅ √18  =  √5 ⋅ 3√2

√5 ⋅ √18  =  3√(5 ⋅ 2)

√5 ⋅ √18  =  3√10

Question 2 :

Simplify : 

³√7 ⋅ ³√8

Solution :

³√7 ⋅ ³√8  =  ³√7 ⋅ ³√(2 ⋅ 2 ⋅ 2)

³√7 ⋅ ³√8  =  ³√7 ⋅ 2

³√7 ⋅ ³√8  =  2³√7

Question 3 :

Simplify : 

3√35 ÷ 2√7

Solution :

3√35 ÷ 2√7  =  3√35 / 2√7

3√35 ÷ 2√7  =  (3 / 2) ⋅ (√35 / √7)

3√35 ÷ 2√7  =  (3 / 2) ⋅ √(35 / 7)

3√35 ÷ 2√7  =  (3 / 2) ⋅ √5

√5 ⋅ √18  =  3√5 / 2

Question 4 :

Simplify : 

³√56 ÷ ³√7

Solution :

³√56 ÷ ³√7  =  ³√(56 ÷ 7)

³√56 ÷ ³√7  =  ³√8

³√56 ÷ ³√7  =  ³√(2 ⋅ 2 ⋅ 2)

³√56 ÷ ³√7  =  2

Question 5 :

Simplify : 

√3(√3 + √12)

Solution : 

√3(√3 + √12)  =  √3 ⋅ √3 + √3 ⋅ √12

√3(√3 + √12)  =  3 + √(3 ⋅ 12)

√3(√3 + √12)  =  3 + √36

√3(√3 + √12)  =  3 + 6

√3(√3 + √12)  =  9

Question 6 :

Simplify : 

(3 - √2)(3 + √2)

Solution : 

Using the algebraic identity a² - b²  =  (a + b)(a - b), 

(3 - √2)(3 + √2)  =  3² - (√2)²

(3 - √2)(3 + √2)  =  9 - 2

(3 - √2)(3 + √2)  =  7

Question 7 :

Simplify : 

√(5/9y⁴)

Solution :

√(5/9y⁴)  =  √5 / √9y⁴

√(5/9y⁴)  =  √5 / √(3y² ⋅ 3y²)

√(5/9y⁴)  =  √5 / 3y²

Question 8 : 

Simplify :  

(√3)³ + √27

Solution :

(√3)³ + √27  =  (√3 ⋅ √3 ⋅ √3) + √(3⋅ 3 ⋅ 3)

(√3)³ + √27  =  (3 ⋅ √3) + 3√3

(√3)³ + √27  =  3√3 + 3√3

(√3)³ + √27  =  6√3

Question 9 :

Simplify the radical expression : 

√243 - 5√12 + √27 

Solution : 

Decompose 243, 12 and 27 into prime factors using synthetic division. 

√243  =  √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)  =  9√3

√12  =  √(2 ⋅ 2 ⋅ 3)  =  2√3

√27  =  √(3 ⋅ 3 ⋅ 3)  =  3√3

So, we have

√243 - 5√12 + √27  =  9√3 - 5(2√3) + 3√3

Simplify.

√243 - 5√12 + √27  =  9√3 - 10√3 + 3√3

√243 - 5√12 + √27  =  2√3

Question 10 :

Simplify :  

(8√3)²

Solution :

(8√3)²  =  8√3 ⋅ 8√3

(8√3)²  =  (8 ⋅ 8)(√3 ⋅ √3)

(8√3)²  =  (64)(3)

(8√3)²  =  192

Question 11 :

Simplify :

(8√117) ÷ (2√52)

Solution : 

Decompose 117 and 52 into prime factors using synthetic division.

(8√117) ÷ (2√52)  =  8(3√13) ÷ 2(2√13)

(8√117) ÷ (2√52)  =  24√13 ÷ 4√13

(8√117) ÷ (2√52)  =  24√13 / 4√13

(8√117) ÷ (2√52)  =  6

Question 12 :

Simplify : 

1 / (2 + √5)

Solution :

Simplifying the above radical expression is nothing but rationalizing the denominator. 

So, rationalize the denominator. 

Here, the denominator is 2 + √5. 

In the given fraction, multiply both numerator and denominator by the conjugate of 2 + √5. That is 2 - √5.

Question 13 :

Simplify : 

2 / √3

Solution :

Simplifying the above radical expression is nothing but rationalizing the denominator. 

So, rationalize the denominator. 

Here, the denominator is √3. 

In the given fraction, multiply both numerator and denominator by √3.

2 / √3  =  (2√3) / (√3 ⋅ √3)

2 / √3  =  2√3 / 3

Question 14 :

Simplify : 

1/√2  +  1/√5

Solution :

To add the above two fractions, make the denominators same. 

Least common multiple of √2 and √5 is 

=  √2 ⋅ √5

=  √(2 ⋅ 5)

=  √10

Then, 

1/√2  +  1/√5  =  √5/√10  +  √2/√10

1/√2  +  1/√5  =  (√5 + √2) / √10

To rationalize the denominator on the right side, multiply both numerator and denominator by √10.

1/√2  +  1/√5  =  [(√5+√2)√10] / (√10 ⋅ √10)

1/√2  +  1/√5  =  (√50 + √20) / 10

1/√2  +  1/√5  =  (√(5 ⋅ 5 ⋅ 2) + √2 ⋅2 ⋅ 5) / 10

1/√2  +  1/√5  =  (5√5 + 2√5) / 10

Question 15 :

Simplify :

(6 + √5) / (6 - √5)

Solution :

Simplifying the above radical expression is nothing but rationalizing the denominator. 

So, rationalize the denominator. 

Here, the denominator is 6 - √5. 

In the given fraction, multiply both numerator and denominator by the conjugate of 6 - √5. That is 6 + √5.

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

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

(6 + √5) / (6 - √5)  =  (6 + √5)² / [6² - (√5)²]

(6 + √5) / (6 - √5)  =  [6² + 2(6)(√5) + (√5)²] / (36 - 5)

(6 + √5) / (6 - √5)  =  [36 + 12√5 + 5] / 31

(6 + √5) / (6 - √5)  =  (41 + 12√5) / 31

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