SIMPLIFYING SQUARE ROOTS

Solved Questions

Simplify each the following square root expressions :

Question 1 :

√64 + √196

Answer :

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

√64 + √196 = 8 + 14

= 22

Question 2 :

√40 + √160

Answer :

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

Question 3 :

2√425 - 3√68

Answer :

Decompose 425 and 68 into prime factors using synthetic division.

2√425 - 3√68 :

= 2(5√17) - 3(2√17)

= 10√17 - 6√17

= 4√17

Question 4 :

√243 - 5√12 + √27

Answer :

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

Question 5 :

-√117 - √52

Answer :

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

Question 6 :

(√17)(√51)

Answer :

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

Question 7 :

(√35)(2√15)

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

Question 8 :

(14√117) ÷ (7√52)

Answer :

Decompose 117 and 52 into prime factors using synthetic division.

(14√117) ÷ (7√52) :

= 14(3√13) ÷ 7(2√13)

= 42√13 ÷ 14√13

= 42√13/14√13

= 3

Question 9 :

(7√5)²

Answer :

(7√5)² = 7√5 ⋅ 7√5

= (7 ⋅ 7)(√5 ⋅ √5)

= (49)(5)

= 245

Question 10 :

(√3)³ + √27

Answer :

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

= (3 ⋅ √3) + 3√3

= 3√3 + 3√3

= 6√3

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