What is the difference between \(\sqrt {150} \) and \(\sqrt {54} \) ?

A \(2\sqrt 6 \)

B \(16\sqrt 6 \)

C \(9\sqrt 6 \)

D \(6\sqrt 2 \)

Correct Answer: Option A

Solution:

প্রথমে প্রতিটি সংখ্যাকে মৌলিক সংখ্যার গুণ আকারে প্রকাশ করি ।

\(\sqrt {150} = \sqrt {(3 \times 50)} = \sqrt 3 \times \sqrt {50} = \sqrt 3 \times \sqrt 2 \times \)

\(\sqrt {25} = 5 \times \sqrt 3 \times \sqrt 2 = 5\sqrt 6 \)

\(\sqrt {54} = \sqrt {(2 \times 27)} = \sqrt 2 \times \sqrt {27} = \sqrt 2 \times \sqrt 3 \times \)

\(\sqrt 9 = 3 \times \sqrt 2 \times \sqrt 3 = 3\sqrt 6 \)

So \(\sqrt {150} - \sqrt {54} = 5\sqrt 6 - 3\sqrt 6 = 2\sqrt 6 \)

Shortcut: \(\sqrt {150} - \sqrt {54} = \sqrt {6 \times 25} - \sqrt {6 \times 9} \)

\(\sqrt {6 \times {{(5)}^{2\;}}} - \sqrt {6 \times {{(3)}^{2\;}}} = 5\sqrt 6 - 3\sqrt 6 = 2\sqrt 6 \)