ROOTS OF INTEGERS WORKSHEET

Evaluate each of the following.

1) √4

2) √16

3) √64

4) √144

5)³√27

6) ³√-8

7) ³√512

8) ³√-8000

9) ⁴√64

10) ⁴√81

11) ⁵√32

12) ⁵√243

13) √8

14) √45

15) ³√54

16) √25 + √36

17) √20 + √45

18) 2√18 - √8

19) ³√8 + ³√27

20) √49 + ³√125 + ⁴√256

Simplify the radicals :

21)  √7(6 - √7)

22)  8√7 (4√5 + 6√3)

23)  (5 + √7) (7 - √2)

1. Answer :

Step 1 :

First we write the given number which is inside the radical as product of prime numbers.

√4 = √(2 x 2)

Step 2 :

Since we have two same numbers inside the radical, we can take one term out of the radical.

= 2

2. Answer :

Step 1 :

First we write the number which is inside the radical as product of prime factors.

√16 = √(2 x 2 x 2 x 2)

Step 2 :

For every two 2's, we have to take one 2 out of the square root sign.

= 2 x 2

= 4

3. Answer :

Step 1 :

First we write the number which is inside the radical as product of prime factors.

√64 = √(2 x 2 x 2 x 2 x 2 x 2)

Step 2 :

For every two 2's, we have to take one 2 out of the square root sign.

= 2 x 2 x 2

= 8

4. Answer :

Step 1 :

First we write the number which is inside the radical as product of prime factors.

√144 = √(2 x 2 x 2 x 2 x 3 x 3)

Step 2 :

• For every two 2's, we have to take one 2 out of the square root sign.
• For every two 3's, we have to take one 3 out of the square root sign.

√144 = 2 x 2 x 3

= 12

5. Answer :

Step 1 :

First we write the number which is inside the radical as product of prime factors.

³√27 = √(3 x 3 x 3)

Step 2 :

• Since we have cube root, we have to take one 3 out of the cube root sign.

³√27 = 3

6. Answer :

Writing -8 as product of prime factors.

³√-8 = ³√[(-2)(-2)(-2)]

Since we have three same values which are multiplied inside the cube root sign, we can take one -2 out of the cube root sign.

= -2

7. Answer :

³√512 = ³√(2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2)

For every three same values inside the cube root sign, we can take one value out of the cube root sign.

= 2 x 2 x 2

= 8

8. Answer :

Sicne we have negative sign inside the cube root sign, the answer will also have negative sign.

³√-8000 = ³√[(2 x 2 x 5 x 2 x 2 x 5 x 2 x 2 x 5)

= -(2 x 2 x 5)

= -20

9. Answer :

⁴√64 = ⁴√(2 x 2 x 2 x 2 x 2 x 2)

Sicne we have fourth root, we have to one value out of the fourth root sign

= 2 x 2

⁴√64  = 4

10. Answer :

⁴√81 = ⁴√(3 x 3 x 3 x 3)

Since we have fourth root, we have to one value out of the fourth root sign

= 3

11. Answer :

⁵√32 = ⁵√(2 x 2 x 2 x 2 x 2)

Since we have fifth root, we have to one value out of the fifth root sign

= 2

12. Answer :

⁵√243 = ⁵√(3 x 3 x 3 x 3 x 3)

Since we have fifth root, we have to one value out of the fifth root sign

= 3

13. Answer :

√8 = √(2 x 2 x 2)

= 2√2

14. Answer :

√45 = √(3 x 3 x 5)

We can take one value out of the square root sign.

= 3√5

15. Answer :

³√54 = ³√(3 x 3 x 3 x 2)

= 3³√2

16. Answer :

√25 + √36 = √(5 x 5) + √(6 x 6)

= 5 + 6

= 11

17. Answer :

√20 + √45 = √(2 x 2 x 5) + √(3 x 3 x 5)

= 2√5 + 3√5

= 5√5

18. Answer :

2√18 - √8 = 2√(3 x 3 x 2) - √(2 x 2 x 2)

= 2(3√2) - 2√2

= 6√2 - 2√2

= 4√2

19. Answer :

³√8 + ³√27 = ³√(2 x 2 x 2) + ³√(3 x 3 x 3)

= 2 + 3

= 5

20. Answer :

√49 + ³√125 + ⁴√256 :

= √(7 x 7) + ³√(5 x 5 x 5) + ⁴√(4 x 4 x 4 x 4)

= 7 + 5 + 4

= 16

21. Answer :

√7(6 - √7)

Using distributive property,

= 6√7 - √7√7

= 6√7 - 7

22. Answer :

= 8√7 (4√5 + 6√3)

Using distributive property,

= 8√7 (4√5) + 8√7 (6√3)

= 32√(7 x 5) + 48√(7 x 3)

= 32√35 + 48√21

23. Answer :

(5 + √7) (7 - √2)

Using distributive property,

= (5 + √7) (7 - √2)

= 5(7) - 5√2 + 7√7 - √2(√7)

= 35 - 5√2 + 7√7 - √14

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