Question

If $ 1125 = {3^m} \times {5^n} $ find m and n.

Answer 1

Hint: First of all we will find the prime factors for the term on the left hand side of the equation and then will compare the term with the common base with its power on both the sides of the equation.

Complete step-by-step answer:
Take the given expression: $ 1125 = {3^m} \times {5^n} $
Prime factorization can be defined as the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than $ 1 $ and which are not the product of any two smaller natural numbers. For Example: $ 2,{\text{ 3, 5, 7,}}...... $ $ 2 $ is the prime number as it can have only $ 1 $ factor. Factors are the number $ 1 $ and the number itself.
 $ 1125 = 9 \times 125 = {3^2} \times {5^3} $
Now, the prime factors for the term on the left can be written as –
 $ {3^2} \times {5^3} = {3^m} \times {5^n} $
When bases are common on both the sides of the equation then the powers are compared and then are equal.
 $ \Rightarrow m = 2 $ and $ n = 3 $
This is the required solution.
So, the correct answer is “ $ m = 2 $ and $ n = 3 $ ”.

Note: Prime factorisation can also be found by the factor tree method. It is the factor-tree method which expresses the prime factors of the composite number in the form tree. One can get different factor trees to get the same prime factorization of the same composite number. Be good in multiples and remember it at least twenty. Go through the different power and exponent equation which states that when base is common then powers are equal in any equation.