LAWS OF EXPONENTS WORKSHEET

1) If x^-1/3 = 5/2, then find the value of x.

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

3) If 5^x/25^y = 125, then solve for x in terms of y.

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

5) If a and b are positive even integers, which of the following is greatest ?

A) (-2a)^b

B) (-2a)^2b

C) (2a)^b

D) 2a^2b

6) If √(x√x) = x^a, then find the value of a.

7) If x² = y³ and x^3z = y⁹, then find the value of z.

8) If n³ = x, n⁴ = 20x and n > 0, then find the value of n.

9) If (√9)^-7 ⋅ (√3)^-4 = 3^k, then find the value of k.

10) In the equation 2√(x - 2) = 3√2, if x ≥ 2, then find the value of x.

1. Answer :

x^-1/3 = 5/2

x = (5/2)^-3/1

x = (5/2)^-3

x = (2/5)³

Distribute the exponent to numerator and denominator.

x = 2³ / 5³

x = 8/125

2. Answer :

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

3. Answer :

5^x/25^y = 125

5^x/(5²)^y = 5³

5^x/5^2y = 5³

5^{x - 2y} = 5³

x - 2y = 3

x = 2y + 3

4. Answer :

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

5. Answer :

Because a and b are positive even integers, better we can assume some values for a and b and go through each choice.

Let a = 2 and b = 2.

Substitute a = 2 and b = 2 in each option.

A : [-2(2)]² = (-4)² = 16

B : [-2(2)]²⁽²⁾ = (-4)⁴ = 256

C : [2(2)]² = (4)² = 16

D : 2(2)²⁽²⁾ = 2(2)⁴ = 2(16) = 32

So, option B is the greatest.

6. Answer :

√(x√x)  = x^a

√(x ⋅ x^1/2)  = x^a

√(x^{1 + 1/2})  = x^a

√(x^3/2) = x^a

(x^3/2)^1/2 = x^a

x^3/4 = x^a

3/4 = a

7. Answer :

x^3z = y⁹

x^3z = y³⁽³⁾

x^3z = (y³)³

Substitute x² for y³.

x^3z = (x²)³

x^3z = x⁶

3z = 6

Divide each side by 3.

z = 2

8. Answer :

n⁴ = 20x

n³ ⋅ n = 20x

Substitute x for n³.

x ⋅ n = 20x

nx = 20x

Divide each side by x.

n = 20

9. Answer :

(9^1/2)^-7 ⋅ (3^1/2)^-4 = 3^k

(9)^-7/2 ⋅ (3)^-4/2 = 3^k

(3²)^-7/2 ⋅ 3^-2 = 3^k

3^{2 ⋅ (-7/2)} ⋅ 3^-2 = 3^k

3^-7 ⋅ 3^-2 = 3^k

3^{-7 - 2} = 3^k

3^-9 = 3^k

k = -9

10. Answer :

2√(x - 2) = 3√2

Square both sides to get rid of the radicals.

[2√(x - 2)]² = (3√2)²

2² ⋅ [√(x - 2)]² = 3² ⋅ (√2)²

4 ⋅ (x - 2) = 9 ⋅ 2

4x - 8 = 18

Add 8 to each side.

4x = 26

Divide each side by 4.

x = 6.5

Video Lessons

Introduction to Exponents

Laws of Exponents - 1

Laws of Exponents - 2

Laws of Exponents - 3

Laws of Exponents - 4

Important Stuff

Problem - 1

Problem - 2

Problem - 3

Problem - 4

Problem - 5

Problem - 6

Problem - 7

Problem - 8

Problem - 9

Problem - 10

Problem - 11

Problem - 12

Problem - 13

Problem - 14

Problem - 15

Problem - 16

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