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. 

11. Evaluate √(41 - (√21 + (√(19 - √9)))

12. The square root of 272² - 128² is

a)  144   b)  200    c)  240      d) 256

13. How many two digit numbers satisfy this property : The unit digit of the square of the two digit number is 8 ?

a)  1      b)  2     c)  3    d) none

14.  If 52/x = √(169/289), the value of x is :

a)  52    b)  58   c)  62    d)  68

15.  If 0.13 + p² = 13, then p equals 

a)  0.01     b)  0.1     c)  10    d) 100

16.  √x  / √441 = 0.02, then the value of x is 

a)  0.1764    b)  1.764     c)  1.64   d)  2.64

17.  If √(x - 1)(y + 2) = 7, x and y being positive whole numbers, then the values of x and y respectively are 

a)  8, 5       b)  15, 12     c)  22, 19      d)  none

18. If √(0.04 x 4 x a) = 0.004 x 0.4 x √b, then a/b is 

a)  16 x 10^-3      b)  16 x 10^-4      c)  16 x 10^-5     d)  16 x 10^-6

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

11. Answer : 

√(41 - (√21 + (√(19 - √9)))

Considering the quantity inside the most interior bracket.

(√(19 - √9)) = (√(19 - √(3⋅3))

= (√(19 - 3))

= √16

= √4 ⋅ 4

= 4

12. Answer : 

= √(272² - 128²) 

Using algebraic identity a² - b², we get 

= (a + b) (a - b)

= √(272 + 128) (272 - 128) 

= √400 (144) 

= √20(20) ⋅ 12 ⋅ 12

= 20 ⋅ 12

= 240

So, option c is correct.

13. Answer : 

We should get 8, by multiplying a number by a same number. It is not true, because multiplying 2 and 4 or multiplying 1 and 8 only we will receive 8.

Then, the answer is none (option d)

14. Answer : 

52/x = √(169/289)

52/x = √(13 ⋅ 13)/(17 ⋅ 17)

52/x = 13/17

Doing cross multiplication, we get

52(17) = 13 x

x = 52(17)/13

x = 4(17)

x = 68

15. Answer : 

If 0.13 / p² = 13, then p equals 

0.13 / p² = 13

0.13/13 = p²

p² = 13/1300

p² = 1/100

p = √(1/100)

p = 1/10

p = 0.1

So, option b is correct.

16. Answer : 

√x  / √441 = 0.02

√x  / √(21 ⋅21) = 0.02

√x/21 = 0.02

√x = 0.02(21)

√x = 0.42

x = (0.42)²

x = 0.1764

So, the answer is option a.

17. Answer : 

Given that, 

√(x - 1)(y + 2) = 7

Option a :

x = 8 and y = 5

√(x - 1)(y + 2) = √(8 - 1)(5 + 2)

= √7(7)

= 7

So, option a is correct. The value of x is 8 and y is 5.

18. Answer : 

√(0.04 x 4 x a) = 0.004 x 0.4 x √b

Squaring both sides, we get

(0.04 x 4 x a) = (0.004 x 0.4 x √b)²

(0.04 x 4 x a) = (0.004 x 0.4 x √b)²

(0.04 x 4 x a) = (0.004)² x (0.4)² x b

a/b = (0.004)² x (0.4)² / (0.04 x 4 x a)

a/b = (0.000016 x 0.16) / (0.04 x 4)

a/b = (0.000016 x 0.16) / (0.04 x 4)

= 0.000016

a/b = 16 x 10^-6

So, option d is correct.

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