Question

Find the square root of 1728 in radical form.

Answer 1

Hint:Finding the root in radical form is nothing but splitting or rewriting the given number in the form of product of prime numbers starting from 2 if divisible and if not proceeding for next prime number. Until it is divided completely. Then taking the prime numbers in pairs and writing one number for each of the pairs outside the radical and then taking the product will be the answer.

Complete step by step answer:
We are given with a number 1728. So first we will write the number as a product of prime numbers. Very first starting from2,
\[{\text{1728 = 2}} \times 864\]
Now continue this until we get the all numbers as prime numbers,
\[1728 = 2 \times 2 \times 432\]
Now 432 can be divided by 2 as well as 3 ; we will go for 3 now.
\[1728 = 2 \times 2 \times 3 \times 144\]
Now 144 is the perfect square but we need all prime numbers. So divide again as,
\[1728 = 2 \times 2 \times 3 \times 3 \times 48\]
Again divide 48 by 3,
\[1728 = 2 \times 2 \times 3 \times 3 \times 3 \times 16\]

Now for 16 go for 2,
\[1728 = 2 \times 2 \times 3 \times 3 \times 3 \times 2 \times 8\]
\[\Rightarrow 1728 = 2 \times 2 \times 3 \times 3 \times 3 \times 2 \times 2 \times 4\]
\[\Rightarrow 1728 = 2 \times 2 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2\]
Now we have all the prime numbers with us. Taking the root on both sides,
\[\sqrt {1728} = \sqrt {2 \times 2 \times 3 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2} \]
Now take the pairs and take one of the numbers for each pair. If we observe then we have 3 pairs of 2 and one pair of 3 .So we can write,
\[\sqrt {1728} = 2 \times 2 \times 2 \times 3\sqrt 3 \]
One of the prime numbers of the product that is without a pair is inside the radical.
Now multiply the numbers,
\[\therefore \sqrt {1728} = 24\sqrt 3 \]

Hence,the square root of 1728 in radical form is $24\sqrt 3$.

Note:Students note that we took a number for each pair out because when a square is written inside the root it cancels the power and the number is to the power only 1. So don’t get confused. And the remaining numbers without pairs should be written in root.Also note that in competitive exams when time is very precious don’t go for writing all the prime numbers, write the number in the form of a product of perfect squares so it would be easier and faster.