# EXPONENTS AND RADICALS WORKSHEET

Problem 1 :

Evaluate the following expression when x = 3 and y = 2.

x2y3

Problem 2 :

Evaluate the following expression when x = 4 and y = 2.

x2/y3

Problem 3 :

If 42n + 3 = 8n + 5, then find the value of n.

Problem 4 :

If 2x/2= 23, then find the value x in terms of y.

Problem 5 :

If ax = b, by = c and  cz = a, then find the value of xyz.

Problem 6 :

If k(2x) = 2x+3 - 2x, then solve for k.

Problem 7 :

Evaluate :

3√-8

Problem 8 :

Evaluate :

√64 + √196

Problem 9 :

(√3)3 + √27

Problem 10 :

Solve for x :

1/3√x = 2

Problem 11 :

Simplify :

3√(x6y9)

Problem 12 :

Solve for x :

√(x3 + 56) = 8

1. Solution :

= x2y3

Substitute x = 3 and y = 2.

= (3)2(2)3

= (3 ⋅ 3)(2 ⋅ 2 ⋅ 2)

= (9)(8)

= 72

2. Solution :

= x2/y3

Substitute x = 4 and y = 2.

= 42/23

= (4 ⋅ 4)/(2 ⋅ 2 ⋅ 2)

= 16/8

= 2

3. Solution :

42n + 3 = 8n + 5

(22)2n + 3 = (23)n + 5

22(2n + 3) = 23(n + 5)

Equate the exponents.

2(2n + 3) = 3(n + 5)

4n + 6 = 3n + 15

n = 9

4. Solution :

2x/2= 23

2x - y = 23

x - y = 3

Add y to both sides.

x = y + 3

5. Solution :

Given : ax = b, by = c and  cz = a.

ax = b

Substitute a = cz.

(cz)x = b

czx = b

Substitute c = by.

(by)zx = b

bxyz = b

bxyz = b1

xyz = 1

6. Solution :

k(2x) = 2x+3 - 2x

Using laws of exponents, we have

k(2x) = 2x ⋅ 23 - 2x

k(2x) = 2x ⋅ 8 - 2x

k(2x) = 2x(8 - 1)

k(2x) = 2x(7)

Divide each side by 2x.

k = 7

7. Solution :

3√-8 = 3√(-2 ⋅ -2 ⋅ -2)

= -2

8. Solution :

√64 + √196

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

 √64 = √(8 ⋅ 8)√64 = 8 √196 = √(14 ⋅ 14)√196 = 14

√64 + √196 = 8 + 14

= 22

9. Answer :

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

= (3  √3) + 3√3

= 3√3 + 3√3

10. Solution :

1/3√x = 2

1/x1/3 = 2

x-1/3 = 2

x = 2-3

x = 1/23

x = 1/8

11. Solution :

= 3√(x6y9)

= (x6y9)1/3

= (x6)1/3(y9)1/3

= x6/3y9/3

= x2y3

12. Solution :

√(x3 + 56) = 8

Take square on both sides.

[√(x3 + 56)]2 = 82

x3 + 56 = 64

Subtract 56 from both sides.

x3 = 8

x3 = 23

x = 2

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