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)