Question

Given $ 270 = 2 \times {3^n} \times 5 $ . Find the value of n.

Answer 1

Hint: In this question the prime factors of the number are given. Prime numbers are the numbers that are only divided by either 1 or by itself and no other number. Prime factors of a number are the multiplication of the prime numbers which will give the original number itself. In this case the prime factors of the number 270 are given in the form of prime numbers 2, 3 and 5. We have to determine the number of times the prime number 3 is repeated in the prime factor of the number 270.

Complete step-by-step answer:
Given:
The number given is 270.
Also, in the question the factors of the number 270 are given as –
$\Rightarrow 270 = 2 \times {3^n} \times 5 $
Where, n is the number of times the prime number 3 is repeated.
Now solving this equation for the value of n we have,
 $
\Rightarrow 270 = 2 \times {3^n} \times 5\\
\Rightarrow {3^n} = \dfrac{{270}}{{2 \times 5}}\\
\Rightarrow {3^n} = 27
 $
We can write 27 as, the cube of 3, so we have,
 $
\Rightarrow {3^n} = 3 \times 3 \times 3\\
\Rightarrow {3^n} = {3^3}
 $
Since the base of both the numbers is same, so now using the property of the exponents, given below,
 $
\Rightarrow {x^m} = {x^n}\\
m = n
 $
So, by comparison we have the value of n as,
 $\Rightarrow n = 3 $
So, in the prime factors of the number 270 the prime factor 3 is repeated 3 times.
Therefore, the value of n is 3.

Note: The alternate method of finding the value of n is by using the “Prime Factorization Method”. In this method first we find the factors of the given number in the form of prime numbers and then by comparing these prime factors with the factors given in the question we can easily calculate the number of times a prime number is repeated.