Question

What is the square root of 105 ?

Answer 1

Hint: From the given question we have been asked the square root. Since the given number \[105\] is not a perfect square, we will find its factors. Then, we will write the given number as a product of its factors. We will check if there is any perfect square among the factors. We will use the identity \[\sqrt{ab}=\sqrt{a}\sqrt{b}\]. If there is no factor that is a perfect square, we cannot express the given number in radical form.

Complete step by step solution:
We are asked to find the square root of this number. Let us consider the given number, \[105\].
We know that the given number is not a perfect square. We need to factorize the given number in order to simplify the given number to a further simplified radical form.
We know that \[105=5\times 3\times 7\].
So, we will get \[\sqrt{105}=\sqrt{5\times 3\times 7}\].
Now, we can use the identity given by \[\sqrt{ab}=\sqrt{a}\sqrt{b}\].
When we use the above identity, we will get \[\sqrt{105}=\sqrt{5}\times \sqrt{3}\times \sqrt{7}\]
We know that the numbers \[3,5,7\] are prime numbers and they are not perfect squares of any number. so, we cannot factorize \[105\] as a multiple of a perfect square. So, let us leave it as it is or as a product of \[\sqrt{5}\times \sqrt{3}\times \sqrt{7}\].
So, we will get the square root of the given number as \[\sqrt{105}=\sqrt{5}\times \sqrt{3}\times \sqrt{7}\] or \[\sqrt{105}\].

Note: Students must be very careful in doing the calculations. Students must have good knowledge in the concept of the square root of a number. What we have found is the simplified form of the square root of the number \[105\] in the radical form. When we write a number under the radical sign, that form of the number is called the radical form.