**Simplifying Square Roots :**

In this section, we are going to learn how to simplify square roots.

Thew following steps will be useful to simplify any square root.

(i) Decompose the number inside the square root into prime factors.

(ii) Inside the square root, if the same number is repeated twice with multiplication, it can be taken out of the square root.

(iii) Combine the like square root terms using mathematical operations.

Example :

√27 + √9 - √12 = √(3 ⋅ 3 ⋅ 3) + √(3 ⋅ 3) - √(2 ⋅ 2 ⋅ 3)

√27 + √9 - √12 = 3√3 + √3 - 2√3

√27 + √9 - √12 = 2√3

**Example 1 : **

Simplify the following square root expression :

√64 + √196

**Solution : **

Decompose 64 and 196 into prime factors using synthetic division.

√64 = √(8 ⋅ 8) √64 = 8 |
√196 = √(14 ⋅ 14) √196 = 14 |

So, we have

√64 + √196 = 8 + 14

√64 + √196 = 22

**Example 2 : **

Simplify the following square root expression :

√40 + √160

**Solution : **

Decompose 40 and 160 into prime factors using synthetic division.

√40 = √(2 ⋅ 2 ⋅ 2 ⋅ 5) = 2√10

√160 = √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5) = 4√10

So, we have

√40 + √160 = 2√10 + 4√10

√40 + √160 = 6√10

**Example 3 : **

**Simplify the following square root expression :**

2√425 - 3√68

**Solution : **

Decompose 425 and 68 into prime factors using synthetic division.

√425 = √(5 ⋅ 5 ⋅ 17) √425 = 5√17 |
√68 = √(2 ⋅ 2 ⋅ 17) √68 = 2√17 |

So, we have

2√425 - 3√68 = 2(5√17) - 3(2√17)

2√425 - 3√68 = 10√17 - 6√17

2√425 - 3√68 = 4√17

**Example 4 : **

Simplify the following square root 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

**Example 5 : **

Simplify the following square root expression :

-√117 - √52

**Solution : **

Decompose 117 and 52 into prime factors using synthetic division.

√117 = √(3 ⋅ 3 ⋅ 13) = 3√13

√52 = √(2 ⋅ 2 ⋅ 13) = 2√13

So, we have

-√117 - √52 = -3√13 - 2√13

-√117 - √52 = -5√13

**Example 6 : **

Simplify the following square root expression :

(√17)(√51)

**Solution : **

Decompose 17 and 51 into prime factors.

Because 17 is a prime number, it can't be decomposed anymore. So, √17 has to be kept as it is.

√51 = √(3 ⋅ 17) = √3 ⋅ √17

So, we have

(√17)(√51) = (√17)(√3 ⋅ √17)

(√17)(√51) = (√17 ⋅ √17)√3

(√17)(√51) = 17√3

**Example 7 : **

Simplify the following square root expression :

(√35)(2√15)

**Solution : **

Decompose 35 and 15 into prime factors.

√35 = √(5 ⋅ 7) = √5 ⋅ √7

√15 = √(5 ⋅ 3) = √5 ⋅ √3

So, we have

(√35)(2√15) = ( √5 ⋅ √7) ⋅ 2(√5 ⋅ √3)

(√35)(2√15) = 2(√5 ⋅ √5)(√7 ⋅ √3)

(√35)(2√15) = 2(5)(√(7 ⋅ 3)

(√35)(2√15) = 10√21

**Example 8 : **

Simplify the following square root expression :

(14√117) ÷ (7√52)

**Solution : **

Decompose 117 and 52 into prime factors using synthetic division.

√117 = √(3 ⋅ 3 ⋅ 13) √117 = 3√13 |
√52 = √(2 ⋅ 2 ⋅ 13) √52 = 2√13 |

(14√117) ÷ (7√52) = 14(3√13) ÷ 7(2√13)

(14√117) ÷ (7√52) = 42√13 ÷ 14√13

(14√117) ÷ (7√52) = 42√13 / 14√13

(14√117) ÷ (7√52) = 3

**Example 9 : **

Simplify the following square root expression :

(7√5)^{2}

**Solution :**

**(7√5) ^{2 }= 7**

**(7√5) ^{2 }= (7**

**(7√5) ^{2 }= (49**

**(7√5) ^{2 }= 245**

**Example 10 : **

Simplify the following square root expression :

(√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**

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