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How do we factorize indices and solve the problem? Quite a number of students find it as one of the most difficult steps to understand. For example finding the value of n when the equation given is 3ⁿ⁻¹ + 3ⁿ = 12

Let us examine the equation 3ⁿ⁻¹ + 3ⁿ = 12. Since the equation involves method of addition, it can’t be solved by applying the normal law of indices. Now our job is to make the law applicable and the only solution is by way of factorisation.

First step is by finding the common factor of the 2 addition number, which in this case is 3ⁿ ( being the smallest factor between 3ⁿ⁻¹ + 3ⁿ ). As such, our new equation shall reads as follow :

3ⁿ ( 3⁻¹ + 1 ) = 12

The next step is to solve the numbers in bracket; 3⁻¹ + 1 = 1/3 + 3/3 = 4/3 and our equation shall reads as follow :

3ⁿ ( 4/3 ) = 12

Now we shall solve the entire equation, that is 3ⁿ = 12 x 3/4, thus 3ⁿ = 9

Since 3ⁿ = 9; the final solution could be done by way of ordinary indices or by applying the law of logarithms.

1st Method

3ⁿ = 9 is also the same meaning with 3ⁿ = 3²; therefore n = 2

Under certain circumstances where such a method can’t be used, the following shall method shall be applied.

2nd Method

3ⁿ = 9

n = log₃ 9 = log₃ 3² = 2 log₃ 3

Since log₃ 3 = 1, therefore n = 2