Question

What is the value of 4 times the square root of 3 divided by the square root of 2?

Answer 1

Hint: Analyze the words in the question statement and convert them into a mathematical expression. When you have any root (square root for example) within the denominator of a fraction you'll "remove" it multiplying and dividing the fraction for the identical quantity. The concept is to avoid any irrational number within the denominator.

Complete step by step answer:
Let us first separate the question statement to parts for easy understanding. The statement can be divided into 2 parts connected by 1 joiner.
Phrase 1: “4 times the square root of 3” $ = \;4(\sqrt 3 )$
Phrase 2: “Square root of 2” $ = \;\sqrt 2 $
These 2 phrases are connected by “divided”, which means there is a division function in between.
The final expression is $\dfrac{{\;4(\sqrt 3 )}}{{\sqrt 2 }}$.
Now to solve the above expression, rationalisation is required as we can clearly see that there is an irrational number in the denominator of the expression. To rationalize the denominator, we need to multiply both the numerator and the denominator with the same number in the denominator, again.
$\dfrac{{\;4(\sqrt 3 )}}{{\sqrt 2 }} = \dfrac{{\;4(\sqrt 3 ) \times \sqrt 2 }}{{\sqrt 2 \times \sqrt 2 }}$
$\Rightarrow \dfrac{{\;4(\sqrt 3 )}}{{\sqrt 2 }} = \dfrac{{4 \times \sqrt 6 }}{{{{(\sqrt 2 )}^2}}} \\
\Rightarrow\dfrac{{\;4(\sqrt 3 )}}{{\sqrt 2 }} = \dfrac{{4 \times \sqrt 6 }}{2} \\
\therefore\dfrac{{\;4(\sqrt 3 )}}{{\sqrt 2 }} = 2\sqrt 6 $

Thus, $2\sqrt 6 $ is the final answer.

Note: In elementary algebra, root rationalisation is defined as a process by which radicals within the denominator of an algebraic fraction are eliminated. This technique could also be extended to any algebraic denominator, by multiplying the numerator and therefore the denominator by all algebraic conjugates of the denominator, and expanding the new denominator into the norm of the old denominator. However, except in special cases, the resulting fractions may have huge numerators and denominators.