Question

Identify whether \[\sqrt{32}\] is rational or irrational?

Answer 1

Hint: In the given question, we are given a square root of a number and we have to tell if it is rational or irrational. We will first write the factors of the number inside the square root, then we will take out the terms which have frequency 2. And then, we will have the expression as, \[4\sqrt{2}\]. Then, we will see the terms if they are rational or not. We know that 4 is rational and \[\sqrt{2}\] is irrational and we know that the product of rational and irrational is always irrational. Hence, we will have the result.

Complete step by step solution:
According to the given question, we are given a square root of a number, which is, 32. We are asked to find if the given root gives a rational or irrational number or simply whether the root is rational or irrational.
Rational number refers to those numbers that can be represented in the form of ratio having a numerator and a denominator. For example - \[0.5\] is a rational number as it can also be written as, \[\dfrac{1}{2}\].
An irrational number refers to those numbers which cannot be expressed as a fraction or in other words we can say, they are unending numbers.
For example – the value of \[\pi =3.142...\]
The expression we have is,
\[\sqrt{32}\]
We will first write its factors, so we have,
\[= \sqrt{4\times 8}\]
We can take 4 out as \[4=2\times 2\], so we get,
\[= 2\sqrt{8}\]
We will now write the factors of 8 and so we get,
\[= 2\sqrt{2\times 4}\]
Again we will take 4 out, we will get the expression as,
\[= 2\times 2\sqrt{2}\]
Multiplying the terms further, we get,
\[= 4\sqrt{2}\]
The above expression is in the most reduced form. We can see there are terms in it 4 and \[\sqrt{2}\].
We know that 4 is a rational number but \[\sqrt{2}\] is not a rational number, it is an irrational number.
And the product of a rational and an irrational number is always irrational.
Therefore, \[\sqrt{32}\] is an irrational number.

Note: Do not directly start finding the square root of the given rather reduce it first. Then, reflect on the terms you will get in the reduced expression and accordingly write whether the given number is rational or irrational. Also, the clear idea of rational and irrational numbers should be known.