PROPERTIES OF RADICALS WORKSHEET

1. Simplify : √6 ⋅ √15

2. Simplify : √35 ÷ √7

3. Simplify : 3√425 + 4√68

4. Simplify : √243 - 5√12 + √27

5. Simplify : √4 + ³√27 + ⁴√64

6. Simplify : ³√4 ⋅ ³√16

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

8. Solve for x : 2√x - 2 = 10. 

9. If ³√a = 1/2, then find the value of a. 

10. If (³√8)⁷ ⋅ (√2)^-4 = 2^k, then solve for k. 

1. Answer :

= √6 ⋅ √15

= √(6 ⋅ 15)

= √(2 ⋅ 3 ⋅ 3 ⋅ 5)

= 3√(2 ⋅ 5)

= 3√10

2. Answer :

= √35 ÷ √7

= √(35/7)

= √5

3. 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

4. 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

5. Answer : 

√4 = √(2 ⋅ 2) = 2

³√27 = ³√(3 ⋅ 3 ⋅ 3) = 3

⁴√625 = ⁴√(5 ⋅ 5 ⋅ 5 ⋅ 5) = 5

√4 + ³√27 + + ⁴√64 :

= 2 + 3 + 5

= 10

6. Answer : 

= ³√4 ⋅ ³√16

= ³√(4 ⋅ 16)

= ³√(4 ⋅ 4 ⋅ 4)

= 4

7. Answer : 

√y = 1/5

y = (1/5)²

y = 1²/5²

y = 1/25

8. Answer : 

2√x - 2 = 10

2√x = 12

√x = 6

x = 6²

x = 36

9. Answer : 

³√a = 1/2

a = (1/2)³

a = 1³/2³

a = 1/8

10. Answer : 

(³√8)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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