Question

Classify the following number as rational or irrational $\sqrt{441}$.

Answer 1

Hint: In this question, we have to classify $\sqrt{441}$ as rational or irrational numbers. For this, we will understand what are rational and irrational numbers. After that, we will factorize 441 and try to find the square root of 441. From that, we will check if the given number is rational or irrational. We will use $\sqrt{a\times a\times b\times b}=a\times b$ as $\sqrt{a\times a\times b\times b}=\sqrt{{{a}^{2}}\times {{b}^{2}}}=\sqrt{\left( a{{b}^{2}} \right)}=a\times b$.

Complete step by step solution:
Here, we are given the number as $\sqrt{441}$ and we have to check if it is a rational or irrational number. Let us first understand the meaning of rational and irrational numbers. Rational numbers are the numbers, which can be expressed in the form of $\dfrac{p}{q}$ where $q\ne 0$. Thus, rational numbers can be fractional, decimal, or integer. Irrational numbers are the numbers, which cannot be expressed in the form of $\dfrac{p}{q}$. Hence, these numbers can never be expressed in fractional form. In decimal form, these numbers are non-terminating and non-recurring. For example,$\sqrt{2}$. is an irrational number as it cannot be expressed in fractional form.
Now to check if $\sqrt{441}$ is rational or not, let us first factorize 441. We get: $441=7\times 7\times 3\times 3$.
Therefore, $\Rightarrow \sqrt{441}=\sqrt{7\times 7\times 3\times 3}=\sqrt{{{\left( 7\times 3 \right)}^{2}}}=7\times 3=21$.
Since 441 is the perfect square of 21.
So, $\sqrt{441}=21$ and 21 can be expressed as $\dfrac{21}{1}$ which is of $\dfrac{p}{q}$ form where, $q\ne 0$. So, $\sqrt{441}$ is a rational number.

Note: Students should note that, if any of the prime factors was left in the square root, then it would have become an irrational number because all square roots of prime numbers are irrational. For example: $\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11}\ldots \ldots \ldots $ all are irrational numbers and thus product of any rational number with irrational number will also be irrational number.