LAWS OF EXPONENTS WORKSHEET

Problems 1-10 : Simplify each expression using the laws of exponents.

Problem 1 :

2m² ⋅ 2m³

Problem 2 :

m⁴ ⋅ 2m^-3

Problem 3 :

(4a³)²

Problem 4 :

(x³)⁰

Problem 5 :

Problem 6 :

(x²y^-1)⁴

Problem 7 :

(3x)(2x^1/2)

Problem 8 :

Problem 9 :

(6ab²c³)(4b^-2c^-3d)

Problem 10 :

(x^-ay^b)³(x^-3y^-2)^-a

Problem 11 :

Find the value of y :

(y²)^2.5 = -32

Problem 12 :

h(x) = 2^x

The function is h is defined above. What is h(5) - h(3)?

Problem 13 :

If 3^x  = 10, what is the value of 3^x-3?

Problem 14 :

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

Problem 15 :

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

Problem 16 :

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

Problem 17 :

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

Problem 18 :

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

Problem 19 :

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

Problem 20 :

If x²y³ = 10 and x³y² = 8, what is the value of x⁵y⁵?

Problem 21 :

If x⁸y⁷ = 333 and x⁷y⁶ = 3, what is the value of xy?

Problem 22 :

If 2^{x + 3} - 2^x = k(2^x), what is the value of k?

Problem 23 :

If x^ac ⋅ x^bc = x³⁰, x > 1 and a + b = 5, what is the value of c?

Problem 24 :

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

Problem 25 :

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

Problem 26 :

x - 2 = √x

In the equation above x ≥ 0, find all the values of x which satisfy the equation.

Answers

1. Answer :

= 2m² ⋅ 2m³

= (2 ⋅ 2)(m² ⋅ m³)

= 4m^{2 + 3}

= 4m⁵

2. Answer :

= m⁴ ⋅ 2m^-3

= 2(m⁴ ⋅ m^-3)

= 2m^{4 - 3}

= 2m¹

3. Answer :

= (4a³)²  

= 4²(a³)²

= 16a^3 ⋅ 2

= 16a⁶

4. Answer :

= (x³)⁰

Anything to the zeroth power is equal to 1.

= 1

5. Answer :

6. Answer :

= (x²y^-1)⁴ 

= (x²)⁴(y^-1)⁴

= x⁸y^-4

7. Answer :

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

= 6x^{1 + 1/2}

= 6x^3/2

8. Answer :

9. Answer :

= (6ab²c³)(4b^-2c^-3d)

= 24ab^{2 - 2}c^{3 - 3}d

= 24ab⁰c⁰d

= 24a(1)(1)d

= 24ad

10. Answer :

= (x^-ay^b)³(x^-3y^-2)^-a

= (x^-a)³(y^b)³(x^-3)^-a(y^-2)^-a

= x^-3ay^3bx^3ay^2a

= x^{-3a + 3a}y^{3b + 2a}

= x³⁰y^{3b + 2a}

= (1)y^{2a + 3b}

= y^{2a + 3b}

11. Answer :

(y²)^2.5 = -32

y^{2 x 2.5}= (-2)⁵

y⁵ = (-2)⁵

Since there is same exponent on both sides, bases can be equated.

y = -2

12. Answer :

h(x) = 2^x

h(5) - h(3) :

= 2⁵ - 2³

= (2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2) - (2 ⋅ 2 ⋅ 2)

= 32 - 8

= 24

13. Answer :

3^x-3 = 3^x/3³

= 3^x/(3 ⋅ 3 ⋅ 3)

= 3^x/27

Substitute 3^x = 10. 

= 10/27

14. 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

15. Answer :

5^x/25^y = 125

5^x/(5²)^y = 5³

5^x/5^2y = 5³

5^{x - 2y} = 5³

Since there is same base on both sides, exponents can be equated.

x - 2y = 3

x = 2y + 3

16. 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.

17. 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

18. 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

19. Answer :

n⁴ = 20x

n³ ⋅ n = 20x

Substitute x for n³.

x ⋅ n = 20x

nx = 20x

Divide each side by x.

n = 20

20. Answer :

x²y³ = 10 ----(1)

x³y² = 8 ----(2)

Multiply (1) and (2).

x²y³ ⋅ x³y² = 10 ⋅ 8

x^{2+ 3}y^{3 + 2} = 80

x⁵y⁵ = 80

21. Answer :

x⁸y⁷ = 333 ----(1)

x⁷y⁶ = 3 ----(2)

Divide (1) and (2).

(x⁸y⁷)/(x⁷y⁶) = 333/3

x^{8 - 7}y^7- 6 = 111

xy = 111

22. Answer :

2^{x + 3} - 2^x = k(2^x)

2^x ⋅ 2³ - 2^x = k(2^x)

Factor 2^x on the left side.

2^x(2³ - 1) = k(2^x)

Divide both sides by 2^x.

2³ - 1 = k

8 - 1 = k

7 =  k

23. Answer :

x^ac ⋅ x^bc = x³⁰

x^{ac + bc} = x³⁰

Since there is same base (x) on both sides, exponents can be equated.

ac + bc = 30

c(a + b) = 30

Substitute a + b = 5.

c(5) = 30

Divide both sides by 5.

c = 6

24. 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

25. 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

26. Answer :

x - 2 = √x

Square both sides to get rid of the square root on the right side.

(x - 2)² = (√x)²

(x - 2)(x - 2) = x

x² - 2x - 2x + 4 = x

x² - 4x + 4 = x

Subtrtact x from both sides.

x² - 5x + 4 = 0

Factor and solve.

x² - 4x - x + 4 = 0

x(x - 4) - 1(x - 4) = 0

(x - 4)(x - 1) = 0

x - 4 = 0  or  x - 1 = 0

x = 4  or  x = 1

Verify the solutions with the original equation.

x = 2 staisfies the original squation and x = 1 doesn't.

Therefore, the solution is x = 2.

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