SOLVING EQUATIONS WITH VARIABLE POWER

Solve each of the following equation.

Example 1 :

4^2x+3 = 1

Solution :

4^2x+3 = 1

Using the property a⁰ = 1, we get

4^2x+3 = 4⁰

Since we have same bases on both sides, we can equate the powers.

2x+3 = 0

2x = -3

x = -3/2

Example 2 :

5^3-2x = 5^-x

Solution :

5^3-2x = 5^-x

3-2x = -x

-2x+x = -3

-x = -3

x = 3

Example 3 :

3^1-2x = 243

Solution :

3^1-2x = 243

243 = 3⁵

3^1-2x = 3⁵

Equating the powers, we get

1-2x = 5

1-5 = 2x

2x = -4

x = -2

Example 4 :

3^2a = 3^-a

Solution :

3^2a = 3^-a

Equating the powers, we get

2a = -a

2a+a = 0

3a = 0

a = 0/3

a = 0

Example 5 :

4^3x-2 = 1

Solution :

4^3x-2 = 1

Using the property 4⁰.

4^3x-2 = 4⁰

3x-2 = 0

3x = 2

x = 2/3

Example 6 :

4^2p = 4^-2p-1

Solution :

Equating the powers, we get

2p = -2p-1

2p+2p = -1

4p = -1

p = -1/4

Example 7 :

6^-2a = 6^2-3a

Solution :

6^-2a = 6^2-3a

Equating the powers, we get

-2a = 2-3a

-2a+3a = 2

a = 2

Example 8 :

2^2x+2 = 2^3x

Solution :

2^2x+2 = 2^3x

Equating the powers, we get

2x+2 = 3x

2x-3x = -2

-x = -2

x = 2

Example 9 :

6^3m ⋅ 6^-m = 6^-2m

Solution :

6^3m ⋅6^-m = 6^-2m

6^3m-m = 6^-2m

By equating the powers, we get

2m = -2m

2m + 2m = 0

4m = 0

m = 0/4

m = 0

Example 10 :

2^x/2^x = 2^-2x

Solution :

2^x/2^x = 2^-2x

2^x (2^-x) = 2^-2x

2^x-x = 2^-2x

2⁰ = 2^-2x

-2x = 0

x = 0/2

x = 0

Example 11 :

10^-3x ⋅ 10^x = 1/10

Solution :

10^-3x ⋅ 10^x = 1/10

By using one of the properties in exponents, we get

10^-3x ⋅ 10^x = 10^-1

-3x+x = -1

-2x = -1

x = 1/2

Example 12 :

3^-2x+1 ⋅ 3^-2x-3 = 3^-x

Solution :

3^-2x+1 ⋅ 3^-2x-3 = 3^-x

3^-2x+1-2x-3 = 3^-x

3^-4x-2 = 3^-x

-4x-2 = -x

-4x+x = 2

-3x = 2

x = -2/3

Example 13 :

Consider the equation 9^m/9ⁿ = 9²

a. Find two numbers m and n that satisfy the equation.

b. Describe the number of solutions that satisfy the equation. Explain your reasoning.

Solution :

Given that, 9^m/9ⁿ = 9²

9^{m - n} = 9²

Since the bases area equal, we can equate the powers.

m - n = 2

a) Two possible values of m and n to satisfy the above equation is 

m = 5 and n = 3 or m = 3 and n = 1

b) There may be infinite number of solutions.

Example 14 :

Find the value of x that makes 8^3x/8^{2x + 1} = 8⁹ true. Explain how you four your answer.

Solution :

Given that 8^3x/8^{2x + 1} = 8⁹

Using rules in exponents,

8^{3x - (2x + 1)} = 8⁹

Distributing the negative sign and combine the like terms, we get

8^{3x - 2x - 1} = 8⁹

8^{x - 1} = 8⁹

Since the bases are equal, we can equate the powers. We get

x - 1 = 9

x = 9 + 1 

x = 10

So, the value of x is 10.

Example 15 :

Which of the following is equivalent to

7^x ⋅ x⁷ / 7⁷ ⋅ x^x  ?

a)  1      b)  (x - 7)^7/x     c)  (x/7)^{x - 7}   d)  (7/x)^{x - 7}

Solution :

Given that, 7^x ⋅ x⁷ / 7⁷ ⋅ x^x 

Using quotient rule,

= 7^{x - 7} ⋅ x^{7 - x}

= 7^{x - 7} ⋅ x^{-(x - 7)}

To convert the negative exponent as positive exponent, we get

= 7^{x - 7} / x^{(x - 7)}

Since we have same powers for both numerator and denominator, we get can use only one power for both numerator and denominator.

= (7/x)^{x - 7} 

So, option d is correct.

Example 16 :

If 

(3ab²) ⋅ (2a²  b)³  / 8a² b²  = 3a^mbⁿ ?

what is the value of m + n ?

Solution :

(3ab²) ⋅ (2a²  b)³  / 8a² b²  = 3a^mbⁿ 

(3ab²) ⋅ (2)³ (a²)³  (b)³  / 8a² b²  = 3a^mbⁿ 

(3ab²) ⋅ 8 a⁶ b³  / 8a² b²  = 3a^mbⁿ 

24 a^{6 + 1} b^{3 + 2} / 8a² b²  = 3a^mbⁿ 

24 a⁷b⁵ / 8a² b²  = 3a^mbⁿ 

3 a^{7 - 2} b^{5 - 2} = 3a^mbⁿ 

3 a⁵ b³ = 3a^mbⁿ 

By comparing the corresponding terms, we get

m = 5 and n = 3

m + n = 5 + 3

= 8

So, the value of m + n is 8.