Question

What is the simplified radical form of \[\sqrt{40}\]?

Answer 1

Hint: In order to find the simplified radical form of \[\sqrt{40}\], firstly we are supposed to find out the largest perfect square from the factors of \[40\]. And then we have to express it in the product of the perfect square and a number such that the product gives the original number. Finally, apply the root and solve it for the perfect square. This will give us the required answer.

Complete step by step answer:
Let us learn about the radical form. Expressing a radical in its simplified form means nothing but expressing the radical in such a form that it cannot be further simplified as square root, cube root or fourth root etc. it also means that removing a radical from the fraction if exists. We simplify the radicals because simplifying the radicals by applying the same simplification rules gives us the same answer which would be easy for tallying.
Now let us express \[\sqrt{40}\] in simplified form.
Firstly, let us find out the factors of 40. They are: \[1,2,4,5,8,10,20,40\].
Now we have to find the largest perfect square from the factors of \[40\]. We have only one perfect square in the list i.e. \[4\], so that would be considered as the largest perfect square.
Now, we will be writing it as the product of perfect squares and a number such that we get \[\sqrt{40}\].
\[\sqrt{40}=\sqrt{4\times 10}\]
Upon separating the terms, we can express it as
\[\Rightarrow \sqrt{40}=\sqrt{4}\times \sqrt{10}\]
Since one of them is a perfect square, we can simplify it. On simplifying, we get-
\[\Rightarrow \sqrt{40}=2\sqrt{10}\]
\[\therefore \] We can observe that this cannot be further simplified. So the simplified radical form of \[\sqrt{40}\] is \[2\sqrt{10}\].

Note: Radicals can be rational numbers but all rational numbers are not radicals. Radicals that have the same root and same radicand are called radicals. Radicals can be used in everyday life in calculating areas, surface areas, total surface areas etc.