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 4^{2n + 3} = 8^{n + 5}, then find the value of n.

Problem 4 :

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

Problem 5 :

If a^x = b, b^y = c and  c^z = a, then find the value of xyz.

Problem 6 :

If k(2^x) = 2^x+3 - 2^x, 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 :

4^{2n + 3} = 8^{n + 5}

(2²)^{2n + 3} = (2³)^{n + 5}

2^{2(2n + 3)} = 2^{3(n + 5)}

Equate the exponents.

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

4n + 6 = 3n + 15

n = 9

4. Solution :

2^x/2^y = 2³

2^{x - y} = 2³

x - y = 3

Add y to both sides.

x = y + 3

5. Solution :

Given : a^x = b, b^y = c and  c^z = a.

a^x = b

Substitute a = c^z.

(c^z)^x = b

c^zx = b

Substitute c = b^y.

(b^y)^zx = b

b^xyz = b

b^xyz = b¹

xyz = 1

6. Solution :

k(2^x) = 2^x+3 - 2^x

Using laws of exponents, we have

k(2^x) = 2^x ⋅ 2³ - 2^x

k(2^x) = 2^x ⋅ 8 - 2^x

k(2^x) = 2^x(8 - 1)

k(2^x) = 2^x(7)

Divide each side by 2^x.

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 + √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/x^1/3 = 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

= x^6/3y^9/3

= 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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