Question

Find the value of \[2\sqrt{54}-6\sqrt{\dfrac{2}{3}}-\sqrt{96}\] .

Answer 1

Hint: To solve this expression, first make the digit inside the square root equal for all the three terms of the expression. \[\sqrt{54}\] can be written as \[3\sqrt{6}\] and \[\sqrt{96}\] can be written as \[4\sqrt{6}\] . We can also write \[6\sqrt{\dfrac{2}{3}}\] as \[\sqrt{24}\] and \[\sqrt{24}\] can be written as \[2\sqrt{6}\] . Now, put these all in the given expression and solve it further.

Complete step-by-step answer:
According to the question, we have to find the value of \[2\sqrt{54}-6\sqrt{\dfrac{2}{3}}-\sqrt{96}\] . Each term of the expression has a different number inside the square root. So, first we have to make the numbers inside the square root equal for each term given in the expression.
The first term that we have is \[2\sqrt{54}\] . Using factors of 54 we can write it as,
\[\begin{align}
  & 2\sqrt{54} \\
 & =2\sqrt{9\times 6} \\
\end{align}\]
9 is a perfect square, so it will come out of the square root as 3.
\[\begin{align}
  & 2\sqrt{9\times 6} \\
 & =2.3\sqrt{6} \\
 & =6\sqrt{6} \\
\end{align}\]
The second term that we have is \[6\sqrt{\dfrac{2}{3}}\] . Taking 6 inside the square root we get
\[\begin{align}
  & 6\sqrt{\dfrac{2}{3}} \\
 & =\sqrt{\dfrac{2}{3}\times 36} \\
 & =\sqrt{24} \\
 & =\sqrt{4\times 6} \\
\end{align}\]
4 is a perfect square, so it will come out of the square root as 2.
\[\begin{align}
  & \sqrt{4\times 6} \\
 & =2\sqrt{6} \\
\end{align}\]
The third term that we have is \[\sqrt{96}\] . Using factors of 96 we can write it as,
\[\begin{align}
  & \sqrt{96} \\
 & =\sqrt{16\times 6} \\
\end{align}\]
16 is a perfect square, so it will come out of the square root as 4.
\[\begin{align}
  & \sqrt{16\times 6} \\
 & =4\sqrt{6} \\
\end{align}\]
Now, transforming the given expression
\[\begin{align}
  & 2\sqrt{54}-6\sqrt{\dfrac{2}{3}}-\sqrt{96} \\
 & =6\sqrt{6}-2\sqrt{6}-4\sqrt{6} \\
 & =6\sqrt{6}-6\sqrt{6} \\
 & =0 \\
\end{align}\]
Hence, the value of the given expression is 0.

Note: In this question, one can think to put the numerical values of each square root term in the expression given. But, if we do so then our calculations will be lengthy and we should also know the value of each square root term which is not easy.