Identify Irrational Number from Options

An irrational number cannot be expressed as a simple fraction \(p/q\), where \(p\) and \(q\) are integers and \(q \neq 0\). We analyze each option to determine if it results in a rational or irrational number.

Option 1: \(4\sqrt{81}\)

• Calculate the square root: \(\sqrt{81} = 9\).
• Perform the multiplication: \(4 \times 9 = 36\).
• Result: $36$ is an integer, hence a rational number (\(36/1\)).

Option 2: \(\sqrt{289}-\sqrt{196}\)

• Calculate the square roots: \(\sqrt{289} = 17\) and \(\sqrt{196} = 14\).
• Perform the subtraction: \(17 - 14 = 3\).
• Result: $3$ is an integer, hence a rational number (\(3/1\)).

Option 3: \(\sqrt{3} \times \sqrt{144}\)

• Calculate the square root: \(\sqrt{144} = 12\).
• Perform the multiplication: \(\sqrt{3} \times 12 = 12\sqrt{3}\).
• Result: \(\sqrt{3}\) is an irrational number. Multiplying it by a non-zero integer ($12$) yields an irrational number (\(12\sqrt{3}\)).

Option 4: \(\sqrt{49}+\sqrt{64}\)

• Calculate the square roots: \(\sqrt{49} = 7\) and \(\sqrt{64} = 8\).
• Perform the addition: \(7 + 8 = 15\).
• Result: $15$ is an integer, hence a rational number (\(15/1\)).

Conclusion: Based on the analysis, the expression \(\sqrt{3} \times \sqrt{144}\) results in an irrational number.