SIMPLIFYING SQUARE ROOTS WORKSHEET

Simplify the following square root expressions :

1)  √64 + √196

2)  √40 + √160

3)  2√425 - 3√68

4)  √243 - 5√12 + √27

5)  -√117 - √52

6)  (√17)(√51)

7)  (√35)(2√15)

8)  (14√117) ÷ (7√52)

9)  (7√5)2

10)  (√3)3 + √27

1. Answer : 

√64 + √196

Because 64 and 196 are perfect squares, we can find the square root of 64 and 194 as shown below. 

√64 = √(8 ⋅ 8)

√64 = 8

√196 = √(14 ⋅ 14)

√196 = 14

√64 + √196 = 8 + 14

= 22

2. Answer : 

√40 + √160

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

√40 + √160 :

= 2√10 + 4√10

= 6√10

3. Answer : 

2√425 - 3√68

Decompose 425 and 68 into prime factors using synthetic division. 

√425 = √(5 ⋅ 5 ⋅ 17)

√425 = 5√17

√68 = √(2 ⋅ 2 ⋅ 17)

√68 = 2√17

2√425 - 3√68 :

= 2(5√17) - 3(2√17)

= 10√17 - 6√17

= 4√17

4. Answer :

√243 - 5√12 + √27

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

√243 - 5√12 + √27 : 

= 9√3 - 5(2√3) + 3√3

= 9√3 - 10√3 + 3√3

= 2√3

5. Answer : 

-√117 - √52

Decompose 117 and 52 into prime factors using synthetic division. 

√117 = √(3 ⋅ 3 ⋅ 13) = 3√13

√52 = √(2 ⋅ 2 ⋅ 13) = 2√13

-√117 - √52 :

= -3√13 - 2√13

= -5√13

6. Answer : 

(√17)(√51)

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

(√17)(√51) : 

(√17)(√3 ⋅ √17)

= (√17 ⋅ √17)√3

17√3

7. Answer :

(√35)(2√15)

Decompose 35 and 15 into prime factors.

√35 = √(5 ⋅ 7) = √5 ⋅ √7

√15 = √(5 ⋅ 3) = √5 ⋅ √3

(√35)(2√15) : 

(√5 ⋅ √7⋅ 2(√5 ⋅ √3)

= 2(√5 ⋅ √5)(√7 ⋅ √3)

= 2(5)(√(7 ⋅ 3)

= 10√21

8. Answer : 

(14√117) ÷ (7√52)

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)

= 42√13 ÷ 14√13

= 42√13/14√13

= 3

9. Answer :

(7√5)= 7√5 ⋅ 7√5

= (7 ⋅ 7)(√5 ⋅ √5)

= (49)(5)

= 245

10. Answer :

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

= (3  √3) + 3√3

= 3√3 + 3√3

= 6√3

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