Question:medium

Which of the following is an irrational number? (A) $2.3$, (B) $\sqrt{13} \times \sqrt{13}$, (C) $\sqrt{441}$

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Quick checks:
Terminating decimals → rational
$\sqrt{a} \times \sqrt{a} = a$
Square root of perfect square → rational
Only roots of non-perfect squares are irrational.
Updated On: Feb 26, 2026
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Solution and Explanation

Step 1: Understanding the Question:
The question requires us to identify which of the given numbers is irrational. An irrational number is a number that cannot be expressed as a simple fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \) is not zero.
Step 2: Key Formula or Approach:
We will examine each option to see if it fits the definition of a rational number.

A number is rational if it is a terminating decimal, a repeating decimal, an integer, or can be written as a fraction.
The square root of a perfect square is a rational number.
The product \( \sqrt{a} \times \sqrt{a} \) is equal to \( a \), which is rational if \( a \) is rational.

Step 3: Detailed Explanation:
Let's analyze each option one by one:
(A) 2.3:
This is a terminating decimal. It can be written as a fraction: \[ 2.3 = \frac{23}{10} \] Since it can be expressed in the form \( \frac{p}{q} \), it is a rational number.
(B) \( \sqrt{13} \times \sqrt{13} \):
Using the property \( \sqrt{a} \times \sqrt{a} = a \), we have: \[ \sqrt{13} \times \sqrt{13} = 13 \] The number 13 is an integer, which can be written as \( \frac{13}{1} \). Therefore, it is a rational number.
(C) \( \sqrt{441} \):
We check if 441 is a perfect square. We know that \( 20^2 = 400 \) and \( 21^2 = 441 \). Thus, 441 is a perfect square. \[ \sqrt{441} = 21 \] The number 21 is an integer, which can be written as \( \frac{21}{1} \). Therefore, it is also a rational number.
Step 4: Final Answer:
All the given numbers are rational. Therefore, none of them is an irrational number.
\[ \boxed{\text{None of these}} \]
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