Question

Which of the following is rational number ?

• A. \(\sqrt{8}\)
• B. \(\sqrt{8} - \sqrt{4}\)
• C. \(\sqrt{2} + \sqrt{2}\)
• D. \(\sqrt{63} \times \sqrt{7}\)

Hint:

When multiplying square roots, combine them: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\).
Check if \(63 \times 7\) is a perfect square.
\(63 = 9 \times 7 \implies 63 \times 7 = 9 \times 7^2\).
Since both \(9\) and \(7^2\) are perfect squares, their product must also be a perfect square, making the result a rational number instantly.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question is from the topic of Number Systems.
We need to evaluate each mathematical expression to determine which one results in a rational number.

Step 2: Key Formula or Approach:
A rational number is any number that can be expressed as a fraction \(\frac{p}{q}\) of two integers, where \(q \neq 0\).
An irrational number cannot be expressed in this form. Typically, the square root of a non-perfect square is irrational.
We will simplify each option and examine the result.

Step 3: Detailed Explanation:
Let us evaluate each option:
Option (A): \(\sqrt{8}\)
Simplify the radical:
\[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]
Since \(\sqrt{2}\) is irrational, \(2\sqrt{2}\) is also irrational.
Option (B): \(\sqrt{8} - \sqrt{4}\)
Simplify both terms:
\[ \sqrt{8} - \sqrt{4} = 2\sqrt{2} - 2 \]
Since \(\sqrt{2}\) is irrational, subtracting an integer from it still results in an irrational number.
Option (C): \(\sqrt{2} + \sqrt{2}\)
Add the like terms:
\[ \sqrt{2} + \sqrt{2} = 2\sqrt{2} \]
This is irrational.
Option (D): \(\sqrt{63} \times \sqrt{7}\)
Combine under a single radical:
\[ \sqrt{63} \times \sqrt{7} = \sqrt{63 \times 7} \]
Let us factor \(63\):
\[ 63 = 9 \times 7 = 3^2 \times 7 \]
Substitute this back:
\[ \sqrt{63 \times 7} = \sqrt{9 \times 7 \times 7} = \sqrt{9 \times 49} = \sqrt{441} \]
We know that \(441 = 21^2\). Therefore:
\[ \sqrt{441} = 21 \]
Since \(21\) is an integer, it is a rational number (it can be written as \(\frac{21}{1}\)).

Step 4: Final Answer:
The expression that represents a rational number is \(\sqrt{63} \times \sqrt{7}\).