FACTORING EXPONENTIAL EXPRESSIONS

Factorise the following :

Problem 1 :

3^2x + 3^x

Solution :

= 3^2x + 3^x

= (3^x)² + 3^x

= 3^x ⋅ 3^x + 3^x

Factoring 3^x, we get

= 3^x (3^x + 1)

Problem 2 :

2ⁿ⁺² + 2ⁿ

Solution :

= 2ⁿ⁺² + 2ⁿ

= 2ⁿ ⋅ 2² + 2ⁿ

= 2ⁿ ⋅ 4 + 2ⁿ

Factoring 2ⁿ, we get

= 2ⁿ (4 + 1)

= 5  ⋅ 2ⁿ 

Problem 3 :

4ⁿ + 4³ⁿ

Solution :

= 4ⁿ + 4³ⁿ

= 4ⁿ + (4ⁿ)³

= 4ⁿ (1 +  (4ⁿ)²)

Factoring 4ⁿ, we get

= 4ⁿ (1 +  4²ⁿ)

Problem 4 :

6ⁿ⁺¹ - 6

Solution :

= 6ⁿ⁺¹ - 6

= 6ⁿ  ⋅ 6 - 6

Factoring 6, we get

= 6(6ⁿ - 1)

Problem 5 :

7ⁿ⁺² - 7

Solution :

= 7ⁿ⁺² - 7

= 7ⁿ ⋅ 7²  - 7

Factoring 7, we get

= 7ⁿ ⋅ (7 ⋅ 7)  - 7

= 7 (7ⁿ ⋅ 7 - 1)

= 7 (7ⁿ⁺¹ - 1)

Problem 6 :

3ⁿ⁺² - 9

Solution :

= 3ⁿ⁺² - 9

= 3ⁿ ⋅ 3² - 9

= 3ⁿ ⋅ 9 - 9

Factoring 9, we get

= 9(3ⁿ - 1)

Problem 7 :

5(2ⁿ) + 2ⁿ⁺²

Solution :

= 5(2ⁿ) + 2ⁿ⁺²

= 5(2ⁿ) + 2ⁿ⋅ 2²

Factoring 2ⁿ, we get

= 2ⁿ (5 + 2²)

= 2ⁿ (5 + 4)

= 9 ⋅ 2ⁿ

Problem 8 :

3ⁿ⁺² + 3ⁿ⁺¹ + 3ⁿ

Solution :

= 3ⁿ⁺² + 3ⁿ⁺¹ + 3ⁿ

= 3ⁿ ⋅ 3² + 3ⁿ ⋅ 3¹ + 3ⁿ

Here we see 3ⁿ in common, factoring it out

= 3ⁿ (3² + 3¹ + 1)

= 3ⁿ (9 + 3 + 1)

= 13 ⋅ 3ⁿ 

Problem 9 :

2ⁿ⁺¹ + 3 (2ⁿ) + 2^n-1

Solution :

= 2ⁿ⁺¹ + 3 (2ⁿ) + 2^n-1

= 2ⁿ ⋅ 2¹ + 3 (2ⁿ) + 2ⁿ⋅ 2^-1

= 2ⁿ (2 + 3 + (1/2))

= 2ⁿ (5 + (1/2))

= 2ⁿ (11/2)

= 2ⁿ (11⋅ 2^-1)

=  11 (2ⁿ⋅ 2^-1)

=  11 (2^n-1)

Problem 10 :

Solve 16^2x = 8^3x

Solution :

16^2x = 8^3x

16^2x - 8^3x = 0

16 = 2⁴ and 8 = 2³

(2⁴)^2x - (2³)^3x = 0

2^8x - 2^9x = 0

2^8x (1 - 2^x) = 0

2^8x  = 0 and 1 - 2^x = 0

8x = 2⁰ and -2^x = -1

8x = 1 and 2^x = 1

x = 1/8 and 2^x = 2⁰ 

x = 1/8 and x = 0

So, the values of x are 0 and 1/8.

Problem 11 :

Solve 4^x = 8^{x + 1}

Solution :

4^x = 8^{x + 1}

(2²)^x = (2³)^{x + 1}

2^2x = 2^{3(x + 1)}

2x = 3(x + 1)

2x = 3x + 3

2x - 3x = 3

-x = 3

x = -3

So, teh value of x is -3.

Problem 12 :

Solve 25^{2 - x} = 125^{2x - 4}

Solution :

25^{2 - x} = 125^{2x - 4}

(5²)^{2 - x} = (5³)^{2x - 4}

5^{2(2 - x)} = 5^{3(2x - 4)}

2(2 - x) = 3(2x - 4)

4 - 2x = 6x - 12

-2x - 6x = -12 - 4

-8x = -16

x = 16/8

x = 2

So, the value of x is 2.

Problem 13 :

Solve 2^{x + 2} - 2^x  = 48

Solution :

2^{x + 2} - 2^x  = 48

2^x 2² - 2^x  = 48

Factoring 2^x, we get

2^x (2² - 1)  = 48

2^x (4 - 1)  = 48

2^x (3)  = 48

2^x  = 48/3

2^x  = 16

2^x  = 2⁴

x = 4

So, the value of x is 4.

Problem 14 :

Solve 4^{x + 3} + 4^x  = 260

Solution :

4^{x + 3} + 4^x  = 260

4^x 4³ + 4^x  = 260

4^x (4³ + 1) = 260

4^x (64 + 1) = 260

4^x (65) = 260

4^x  = 260/65

4^x  = 4

4^x  = 4¹

x = 1

So, the value of x is 1.

Problem 15 :

Solve 2^{x + 5} + 2^x  = 1056

Solution :

2^{x + 5} + 2^x  = 1056

2^x 2⁵ + 2^x  = 1056

2^x (2⁵  + 1) = 1056

2^x (32 + 1) = 1056

2^x (33) = 1056

2^x = 1056/33

2^x = 32

2^x = 2⁵

x = 5

So, the value of x is 5.

Problem 16 :

1/256 = 2^{2 - 5x}

Solution :

1/256 = 2^{2 - 5x}

256 = 2⁸

1/2⁸ = 2^{2 - 5x}

2^-8 = 2^{2 - 5x}

-8 = 2 - 5x

-8 - 2 = -5x 

-5x = -10

x = 10/5

x = 2

So, the value of x is 2.

Problem 17 :

Solve 27^x/9^{2x - 1} = 3^{x + 4}

Solution :

27^x/9^{2x - 1} = 3^{x + 4}

(3³)^x(3^-2)^{2x - 1} = 3^{x + 4}

3^3x 3^{-2(2x - 1)} = 3^{x + 4}

3^3x 3^{-4x + 2} = 3^{x + 4}

3^{3x - 4x + 2} = 3^{x + 4}

3^{-x + 2} = 3^{x + 4}

-x + 2 = x + 4

-x-x = 4 - 2

-2x = 2

x = -1

Problem 18 :

Solve 2^2x+1/2^{x - 3} = 4

Solution :

2^2x+1/2^{x - 3} = 4

2^2x+1 2^{-(x - 3)} = 4

2^{2x+1 - (x - 3)} = 4

2^{2x + 1 - x + 3} = 4

2^{2x + 4} = 4

2^{2x + 4} = 2²

2x + 4 = 2

2x = 2 - 4

2x = -2

x = -1

So, the value of x is -1.