Question

Find variable y represents rational number or irrational number: ${{y}^{2}}=9$

Answer 1

Hint: First we discuss the rational number and irrational numbers. Then, we simplify the given equation by taking the square root and find that the answer is a rational number or irrational number.

Complete step by step solution:
We have given that equation ${{y}^{2}}=9$
We have to find that the variable $y$ represents a rational number or irrational number.
We know that when two numbers represented in the form of ratio of the two integers, are called rational numbers. In rational numbers both numerator and denominator are whole numbers. Denominator of a rational number is not equal to zero.
For example $\dfrac{a}{b}$ is rational number, where $b\ne 0$.
$\dfrac{2}{3}$ is a rational number.
Also, we know that the numbers which cannot be represented in the form of fraction are called irrational numbers.
For example \[\sqrt{2,}\sqrt{3}\] are irrational numbers.
Now, the given equation is ${{y}^{2}}=9$.
When we solve further, we get
$\begin{align}
  & y=\sqrt{9} \\
 & y=3 \\
\end{align}$
As we know that $9$ is the square root of $3$.
As we get an integer, it means the variable $y$ represents a rational number.

Note: Square and square root operations with respect to each other are inverse mathematical operations. Square root of any number is written as $\sqrt{{}}$. Square root of a perfect number is always an integer. If we have given \[{{y}^{2}}=\sqrt{3}\] then, it would be an irrational number. In order to get a rational number ${{y}^{2}}=a$, $a$must be a perfect square.