Question

What is the value of \[\dfrac{{\sqrt {162} + \sqrt {108} }}{{\sqrt {72} + \sqrt {48} }}\] ?

Answer 1

Hint: This question is simply related to how do we convert the root or simplify it. Here we will write the root in terms of product of prime numbers and then we will try to take the perfect roots outside. After that we will take any common factors if there and simplify it further. So let’s start!

Complete step-by-step solution:
Given that,
\[\dfrac{{\sqrt {162} + \sqrt {108} }}{{\sqrt {72} + \sqrt {48} }}\]
Now we will write each root in terms of the product of prime numbers.
\[\sqrt {162} = \sqrt {2 \times 81} = \sqrt {2 \times 3 \times 27} = \sqrt {2 \times 3 \times 3 \times 3 \times 3} = 3 \times 3\sqrt 2 = 9\sqrt 2 \]
As we can see in the root above 162 is finally written in root form but in simplified form. We will do this with the remaining three roots also.
\[\sqrt {108} = \sqrt {2 \times 54} = \sqrt {2 \times 2 \times 27} = \sqrt {2 \times 2 \times 3 \times 3 \times 3} = 2 \times 3\sqrt 3 = 6\sqrt 3 \]
\[\sqrt {72} = \sqrt {2 \times 36} = \sqrt {2 \times 2 \times 18} = \sqrt {2 \times 2 \times 2 \times 3 \times 3} = 2 \times 3\sqrt 2 = 6\sqrt 2 \]
\[\sqrt {48} = \sqrt {2 \times 24} = \sqrt {2 \times 2 \times 12} = \sqrt {2 \times 2 \times 2 \times 2 \times 3} = 2 \times 2\sqrt 3 = 4\sqrt 3 \]
Now we will write these simplified forms in the fraction above.
\[ = \dfrac{{9\sqrt 2 + 6\sqrt 3 }}{{6\sqrt 2 + 4\sqrt 3 }}\]
Taking 3 common from the numerator and 2 common form the denominator,
\[ = \dfrac{{3\left( {3\sqrt 2 + 2\sqrt 3 } \right)}}{{2\left( {3\sqrt 2 + 2\sqrt 3 } \right)}}\]
Cancelling the same term,
\[ = \dfrac{3}{2}\]
Thus this is the simplified answer.
\[\dfrac{{\sqrt {162} + \sqrt {108} }}{{\sqrt {72} + \sqrt {48} }} = \dfrac{3}{2}\]

Note: Note that, we need not to write the value of that root but when we write the root inside the number in product of prime numbers form we have used composite numbers also just for the purpose of convenience. If you observe that in the case of 72 we can simply write 36 as the perfect square of 6. But just for the purpose of convenience we have written the answer in detail.
Also note that we cannot add the numbers with different roots. Like in numerator and denominator both have roots but we cannot add them. Numbers with similar roots only can be added.