PROPERTIES OF SQUARE ROOTS WORKSHEET

1. Simplify : √6 ⋅ √3

2. Simplify : 3√8 ⋅ 2√6

3. Simplify : √35 ÷ √7

4. Simplify : 4√27 ÷ 3√3

5. Simplify : 3√425 + 4√68

6. Simplify : √243 - 5√12 + √27 

7. If √a = 1/5, then find the value of a. 

8. If (√4)⁷ ⋅ (√2)^-4 = 2^k, then solve for k. 

1. Answer :

= √6 ⋅ √3

= √(6 ⋅ 3)

= √(2 ⋅ 3 ⋅ 3)

= 3√2

2. Answer :

= 3√8 ⋅ 2√6

= (3 ⋅ 2)(√8 ⋅ √6)

= 6√(8 ⋅ 6)

= 6√(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3)

= 24√6

3. Answer :

= √35 ÷ √7

= √(35/7)

= √5

4. Answer :

= 4√27 ÷ 3√3

= (4/3)√(27/3)

= (4/3)(√9)

= (4/3)(3)

= 4

5. Answer :

Decompose 425 and 68 into prime factors using synthetic division. 

3√425 + 4√68 : 

= 3(5√17) + 4(2√17)

= 15√17 + 8√17

= 23√17

6. 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

7. Answer : 

√a = 1/5

a = (1/5)²

a = 1²/5²

a = 1/25

8. Answer : 

 (√4)⁷ ⋅ (√2)^-4 = 2^k

2⁷ ⋅ (2^1/2)^-4 = 2^k

2⁷ ⋅ 2^-2 = 2^k

2^{7 - 2} = 2^k

2⁵ = 2^k

k = 5

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