Question

How do you evaluate $\sqrt{12}+5\sqrt{8}-7\sqrt{20}$ ?

Answer 1

Hint: For these kinds of questions, we need to know a little bit about radical expressions or surds. Since it is square root, we have to find the numbers which are repeated twice in the root. We can get those numbers outside since ${{\left( {{a}^{2}} \right)}^{\dfrac{1}{2}}}=a$ . After doing so with the entire given expression, we can add or subtract like how we do with normal numbers.

Complete step by step answer:
Now, let us see what numbers are repeated in the square root.
For that, we need to write down all the numbers given under the root as a product of primes numbers but not composite numbers as composite numbers can still be further broken down into prime numbers.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \sqrt{12}+5\sqrt{8}-7\sqrt{20} \\
 & \Rightarrow \sqrt{4\times 3}+5\sqrt{4\times 2}-7\sqrt{5\times 4} \\
\end{align}$
Let us break them down even more until we are left with only primes numbers.
$\begin{align}
  & \Rightarrow \sqrt{12}+5\sqrt{8}-7\sqrt{20} \\
 & \Rightarrow \sqrt{4\times 3}+5\sqrt{4\times 2}-7\sqrt{5\times 4} \\
 & \Rightarrow \sqrt{2\times 2\times 3}+5\sqrt{2\times 2\times 2}-7\sqrt{5\times 2\times 2} \\
 & \Rightarrow \sqrt{{{2}^{2}}\times 3}+5\sqrt{{{2}^{2}}\times 2}-7\sqrt{5\times {{2}^{2}}} \\
\end{align}$
When we apply square root over a number which is already squared, we get the number itself.
$\Rightarrow \sqrt{{{a}^{2}}}={{\left( {{a}^{2}} \right)}^{\dfrac{1}{2}}}=a$
Let us apply the same thing to our expression.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \sqrt{12}+5\sqrt{8}-7\sqrt{20} \\
 & \Rightarrow \sqrt{4\times 3}+5\sqrt{4\times 2}-7\sqrt{5\times 4} \\
 & \Rightarrow \sqrt{2\times 2\times 3}+5\sqrt{2\times 2\times 2}-7\sqrt{5\times 2\times 2} \\
 & \Rightarrow \sqrt{{{2}^{2}}\times 3}+5\sqrt{{{2}^{2}}\times 2}-7\sqrt{5\times {{2}^{2}}} \\
 & \Rightarrow 2\sqrt{3}+10\sqrt{2}-14\sqrt{5} \\
\end{align}$
We can only add those radicals or subtract which have the same number inside their square root or any ${{n}^{th}}$ root.
Since all the three radicals or surds have different numbers under their root, we cannot simplify it further .
$\therefore $ Hence, upon evaluating $\sqrt{12}+5\sqrt{8}-7\sqrt{20}$, we get $2\sqrt{3}+10\sqrt{2}-14\sqrt{5}$.

Note: Radicals or surds is a very important chapter. This concept is grouped together with other tough chapters and a question can be asked. We should remember all the rules of radicals and the formulae as well. We should be very careful while solving as there is a lot of scope of mistake. There can be a lot of confusion while adding, subtracting or multiplying radicals. We should practice more to get clarity and do them quickly.