Question

What is the square root of $420?$

Answer 1

Hint: Since the given number $420$ 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:
Let us consider the given number, $420.$
We are asked to find the square root of this number.
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 $420=4\times 105.$
So, we will get $\sqrt{420}=\sqrt{4\times 105}.$
Now, we can use the identity given by $\sqrt{ab}=\sqrt{a}\sqrt{b}.$
When we use the above identity, we will get $\sqrt{420}=\sqrt{4}\sqrt{105}.$
We know that the number $4$ is a perfect square whereas the number $105$ is not. Also, we cannot further factorize $105$ as a multiple of a perfect square. So, let us leave it as it is.
We know that the square root of $4$ is $2.$ That is $\sqrt{4}=2.$
So, we will get the square root of the given number as $\sqrt{420}=2\sqrt{105}.$
Hence the square root of $420=2\sqrt{105}.$

Note: What we have found is the simplified form of the square root of the number $420$ in the radical form. When we write a number under the radical sign, that form of the number is called the radical form.