LAWS OF EXPONENTS WORKSHEET

Problem 1 :

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

Problem 2 :

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

Problem 3 :

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

Problem 4 :

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

Problem 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

Problem 6 :

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

Problem 7 :

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

Problem 8 :

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

Problem 9 :

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

Problem 10 :

2√(x - 2)  =  3√2

In the equation above, if x ≥ 2, then find the value of x. 

Detailed Answer Key

Problem 1 :

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

Solution : 

a^-1/2  =  5

a  =  5^-2/1

a  =  5^-2

a  =  1/5²

a  =  1/25

Problem 2 :

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

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

Problem 3 :

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

Solution : 

2^x / 2^y  =  2³

2^{x - y}  =  2³

x - y  =  3

x  =  y + 3

Problem 4 :

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

Solution : 

Let

a^x  =  b -----(1)

b^y  =  c -----(2)

c^z  =  a -----(3)

Substitute a  =  c^z in (1).

(1)-----> (c^z)^x  =  b

c^zx  =  b

Substitute c  =  b^y.

(b^y)^zx  =  b

b^xyz  =  b

b^xyz  =  b¹

xyz  =  1

Problem 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

Solution : 

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.

Problem 6 :

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

Solution : 

√(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

So, the value of a is 3/4.

Problem 7 :

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

Solution : 

x^3z  =  y⁹

x^3z  =  y^{3 ⋅ 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

So, the value of z is 2.

Problem 8 :

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

Solution : 

n⁴  =  20x

n³ ⋅ n  =  20x

Substitute x for n³.

  x ⋅ n  =  20x

nx  =  20x

Divide each side by x.

n  =  20

So, the value of n is 20.

Problem 9 :

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

Solution : 

(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

So, the value of k is -9.

Problem 10 :

2√(x - 2)  =  3√2

In the equation above, if x ≥ 2, then find the value of x. 

Solution : 

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

Simplify. 

4x - 8  =  18

Add 8 to each side. 

4x  =  26

Divide each side by 4. 

x  =  6.5

So, the value of x is 6.5

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