Question

How do you simplify \[\sqrt{48}+\sqrt{108}\] ?

Answer 1

Hint: For solving problems such as these, we need to have a thorough knowledge about rational and irrational fractions and factorization, which will help in simplification to a great extent. In case of this above given problem, we need to factorize the numbers inside the square roots in such a way, that we express them as a product of numbers which can be written down as perfect squares. These square numbers on pulling out of the square root, becomes the number itself. Hence solving such problems becomes easy. After this we need to add the two terms algebraically by taking something common to find the resultant simplified answer.

Complete step by step solution:
Now we start of the problem by writing down the factors of the numbers in product form and in such a way that there are numbers which can be expressed as perfect squares,
  \[\begin{align}
  & \sqrt{48}+\sqrt{108} \\
 & =\sqrt{3\times 16}+\sqrt{3\times 4\times 9} \\
\end{align}\]
Now, observing very carefully we can see that there are numbers in both the terms of the equation which are perfect squares of some numbers. Thus expressing them as perfect square numbers, we get,
\[=\sqrt{3\times {{4}^{2}}}+\sqrt{3\times {{2}^{2}}\times {{3}^{2}}}\]
Now, we know that the square root of a perfect square number is the number itself. Applying this concept in our above formed problem we get,
\[=4\sqrt{3}+6\sqrt{3}\]
Now, adding the two terms algebraically we get,
\[=10\sqrt{3}\]
Now, here we can clearly see that the problem cannot be simplified further, hence this will be the answer to our problem.

Note: These types of problems require an in-depth understanding of the theory of square roots and factorization. We must never forget the golden rule that the square root of a perfect square number is the number itself. We must also be very cautious while we are adding two irrational numbers, because if they don’t have the same number within the square root, we will not be able to add them.