Question

Which among the following is a surd?

• A. $\sqrt{11\times 11\times 27\times 27}$
• B. $\sqrt{12\times 16\times 24}$
• C. $\sqrt{11\times 22\times 27\times 6}$
• D. $\sqrt{3\times 7\times 8\times 42}$

Answer 1

The correct option is B $\sqrt{12\times 16\times 24}$

Surds are values inside square roots that cannot be further simplified into whole numbers or integers.

In other words, surds are irrational numbers.

$\sqrt{12\times 16\times 24}=\sqrt{4\times 3\times 4\times 4\times 6\times 4}=\sqrt{4^4\times 3\times 3\times 2}=\sqrt{4^4\times 3^2\times 2}=4\times 4\times 3\sqrt{2}=48\sqrt{2},\text{a surd.}$

$\sqrt{3\times 7\times 8\times 42}=\sqrt{3\times 7\times 4\times 2\times 7\times 6}=\sqrt{7^2\times 4\times 2\times 3\times 3\times 2}=\sqrt{7^2\times 4\times 4\times 3^2}=\sqrt{7^2\times 4^2\times 3^2}=7\times 4\times 3=84,\text{a rational number.}$

$\sqrt{11\times 22\times 27\times 6}=\sqrt{11\times 11\times 2\times 9\times 3\times 3\times 2}=\sqrt{{11}^2\times 2^2\times 9\times 9}=\sqrt{{11}^2\times 2^2\times 9^2}=11\times 2\times 9=198,\text{a rational number.}$

$\sqrt{11\times 11\times 27\times 27}$
$=\sqrt{{11}^2\times {27}^2}$
$=11\times 27$
$=297,\text{a rational number.}$

Hence, the correct answer is $\sqrt{12\times 16\times 24}.$