Question

If a number $P$ is a positive integer but not a perfect square, then $P$ is definitely _________.

• A. an integer
• B. a terminating decimal number
• C. a rational number
• D. an irrational number

Answer 1

The correct option is D an irrational number

Detailed step-by-step solution:
A perfect square is a number which can be represented as a product of two same numbers.Accordingly, a non perfect square is a number which can not be represented as the product of two same numbers.
Let us consider a non perfect square number, $P=48$.
Then,
$\sqrt{P}=\sqrt{48}.$
$\sqrt{P}=\mathrm{6.9282...}$
Here, $\mathrm{6.9282...}$ is a non-terminating non-recurring decimal number, and non-terminating non-recurring numbers are irrational numbers.
Hence, $\sqrt{P}$ will be an irrational number if $\sqrt{P}$ is not a perfect square.
If $\sqrt{P}$ had been $25$ which can be expressed as $5\times 5$, then $\sqrt{P}$ would be an integer. If $\sqrt{P}$ had been $\frac{25}{144}$ which can be expressed as $\frac{5}{12}\times \frac{5}{12}$ , then $\sqrt{P}$ would be a rational number. If $\sqrt{P}$ had been $\frac{25}{144}$ which can be expressed as $\frac{5}{12}\times \frac{5}{12}$ , then $\sqrt{P}$ would be a rational number.
By considering any value of $\sqrt{P}$ as a positive non perfect square integer, $\sqrt{P}$ will always be an irrational number.
So, option A is the correct answer.