Question

How do you solve the square root of \[108\] minus square root of \[27\] ?

Answer 1

Hint: Problems like these are quite easy to solve once we understand the underlying concept behind the problem clearly. Before solving the problem, first of all we need to understand the problem statement and convert the given language into a mathematical form. In this problem the hence formed mathematical form of the problem will be, \[\sqrt{108}-\sqrt{27}\] . Now we need to write all the factors of the numbers \[108\] and \[27\] in their product form and after that we try to represent the numbers in their square forms. Now suppose we have a number inside the square root of the form \[{{a}^{2}}\] , then we take \[a\] common from the root and take it out. We perform this on both the parts of the square root and then evaluate the answer.

Complete step by step solution:
Now we start off with the solution to the given problem by writing the mathematical form of the problem as,
\[\sqrt{108}-\sqrt{27}\]
Now we need to factorise the numbers inside of the square root and then represent it as the product of the factors. Hence we rearrange the form as,
\[\sqrt{2\times 2\times 3\times 3\times 3}-\sqrt{3\times 3\times 3}\]
Now we write this in the power form as,
\[\sqrt{{{2}^{2}}\times {{3}^{2}}\times 3}-\sqrt{{{3}^{2}}\times 3}\]
Now we take common out of the square root to get,
\[\begin{align}
  & 2\times 3\sqrt{3}-3\sqrt{3} \\
 & \Rightarrow 6\sqrt{3}-3\sqrt{3} \\
 & \Rightarrow 3\sqrt{3} \\
\end{align}\]

So the answer to the problem is \[3\sqrt{3}\] .

Note: Such problems require a basic as well as to some extent an advanced knowledge of powers and square roots and factorisation. We must be very careful while decomposing a large number into factors and then representing the number as a product of these factors. We then need to express the numbers in the form of squares so that we are able to take out the integers out of the square root to get our desired result.