Question

How do you simplify \[\sqrt 2 .\sqrt 3 \]?

Answer 1

Hint: In this question, first we will take both the numbers in a single root so that it will be easy to multiply them. then we will multiply both the numbers. The final answer which we get will be written inside the root and it will be an irrational number.

Complete answer:
In the above question, we are given multiplication of two irrational numbers and the final number we will get is also an irrational number.
But first we should know what a rational number is and what an irrational number is.
$ \Rightarrow $ Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction, $\dfrac{p}{q}$ where p and q are integers. The denominator q is not equal to zero \[\left( {q{\text{ }} \ne {\text{ }}0} \right)\]. Also, the decimal expansion of an irrational number is neither terminating nor repeating.
$\pi $ (pi) and $\sqrt 2 $ are examples of an irrational number.
A rational number is a number that is of the form $\dfrac{p}{q}$ where p and q are integers and q is not equal to 0. The set of rational numbers is denoted by Q. In other words, If a number can be expressed as a fraction where both the numerator and the denominator are integers, the number is a rational number.
Now, in the above question
We have,
\[ \Rightarrow \sqrt 2 .\sqrt 3 \]
Now we will take both the terms under the same root.
$ \Rightarrow \sqrt {2 \times 3} $
Now we will multiply both the terms
 $ \Rightarrow \sqrt 6 $
Therefore, $\sqrt 6 $ will be our required answer.

Note: There are some properties of irrational number:
(A) When any irrational numbers are multiplied by any nonzero rational number, their product is an irrational number. For an irrational number x and a rational number y, their product \[xy{\text{ }} = {\text{ }}irrational\].
(B) Irrational numbers consist of non-terminating and non-recurring decimals.
(C) These are real numbers only.
(D) When an irrational and a rational number are added, the result or their sum is an irrational number only. For an irrational number x, and a rational number y, their result, \[x + y{\text{ }} = {\text{ }}an{\text{ }}irrational{\text{ }}number\].