Question

How do I simplify this?
$\sqrt {\dfrac{2}{3}} + \sqrt {\dfrac{3}{2}} + 2\sqrt {\dfrac{1}{{24}}} $

Answer 1

Hint: Here we are asked to simplify the mathematical expression having three terms. All the denominators are different and so, we have to take the LCM (Least common multiple) for the terms and then simplify.

Complete step by step answer:
LCM is the least or the smallest number with which the given numbers are exactly divisible. It is also known as the least common divisor. Here, we will find the prime factors first and accordingly will find LCM.
Find the prime factors of the denominator.
$
  3 = 3 \\
  2 = 2 \\
  24 = 2 \times 2 \times 2 \times 3 \\
 $
Hence, the LCM is $24$
Now, make the denominators as $24$using the property of the equivalent fraction.
$ = \sqrt {\dfrac{2}{3} \times \dfrac{8}{8}} + \sqrt {\dfrac{3}{2} \times \dfrac{{12}}{{12}}} + 2\sqrt {\dfrac{1}{{24}} \times \dfrac{1}{1}} $
Multiply and divide the terms with the same factor keeps the value the same. Simplify the above expression.
$ = \sqrt {\dfrac{{16}}{{24}}} + \sqrt {\dfrac{{36}}{{24}}} + 2\sqrt {\dfrac{1}{{24}}} $
Now, when denominators are the same, add the numerators.
$ = \dfrac{{\sqrt {16} + \sqrt {36} + 2\sqrt 1 }}{{\sqrt {24} }}$
Simplify the numerators finding the square-root of the terms –
$ = \dfrac{{4 + 6 + 2}}{{\sqrt {24} }}$
Simplify adding the terms –
$ = \dfrac{{12}}{{\sqrt {24} }}$
Find the square-root
$ = \dfrac{{12}}{{2\sqrt 6 }}$
Common factors from the numerator and the denominator cancels each other.
$ = \dfrac{6}{{\sqrt 6 }}$
The above expression can be written as –
$ = \dfrac{{\sqrt 6 \times \sqrt 6 }}{{\sqrt 6 }}$
Common factors from the numerator and the denominator cancels each other.
$ = \sqrt 6 $
This is the required solution.

Note: Prime factorization should be done very carefully as sole answer the LCM depends on the prime factors. Prime factorization can be done by using the factor tree method. Be aware about the difference between the HCF and LCM and apply them accordingly. HCF is the greatest or the largest common factor between two or more given numbers whereas the LCM is the least or the smallest number with which the given numbers are exactly divisible.