Question

How do you simplify $3\sqrt{27}+4\sqrt{12}-\sqrt{300}$ ?

Answer 1

Hint: In this question, we have to simplify the given expression. Thus, we will use the least common multiple method and the basic mathematical rule to get the solution. First, we will find the least common multiple of 27, and then we will form a two pair grouping of the multiples of 27. Then, we will take the square root on both sides of the equation to get the value of $\sqrt{27}$ . Similarly, we will find the value of $\sqrt{12}$ and $\sqrt{300}$. Then, we will substitute the value of the square roots in the given expression. Thus, we will use the basic mathematical rules, to get the required result for the solution.

Complete step by step solution:
According to the question, we have to simplify the given expression.
Thus, we will use the least common multiple method and the basic mathematical rules to get the solution.
The expression given to us is $3\sqrt{27}+4\sqrt{12}-\sqrt{300}$ ---------- (1)
First, we will find the least common multiple of 27, we get
$\begin{align}
  & \text{ 3}\left| \!{\underline {\,
  27 \,}} \right. \\
 & \text{ 3}\left| \!{\underline {\,
  9 \,}} \right. \\
 & \text{ 3}\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
$\Rightarrow LCM(27)=3\times 3\times 3$
Now, we will form a two pair grouping of the multiples of 27, we get
$\Rightarrow 27=\underline{3\times 3}\times 3$
Now, taking the square root on both sides in the above equation, we get
$\Rightarrow \sqrt{27}=3\sqrt{3}$ ------- (2)
Similarly, we will find the least common multiple of 12, we get
$\begin{align}
  & \text{ 2}\left| \!{\underline {\,
  12 \,}} \right. \\
 & \text{ 2}\left| \!{\underline {\,
  6 \,}} \right. \\
 & \text{ 3}\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
$\Rightarrow LCM(12)=2\times 2\times 3$
Now, we will form a two pair grouping of the multiples of 12, we get
$\Rightarrow 12=\underline{2\times 2}\times 3$
Now, taking the square root on both sides in the above equation, we get
$\Rightarrow \sqrt{12}=2\sqrt{3}$ --------- (3)
Similarly, we will find the least common multiple of 300, we get
$\begin{align}
  & \text{ 2}\left| \!{\underline {\,
  300 \,}} \right. \\
 & \text{ 2}\left| \!{\underline {\,
  150 \,}} \right. \\
 & \text{ 3}\left| \!{\underline {\,
  75 \,}} \right. \\
 & \text{ 5}\left| \!{\underline {\,
  25 \,}} \right. \\
 & \text{ 5}\left| \!{\underline {\,
  5 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
$\Rightarrow LCM(300)=2\times 2\times 3\times 5\times 5$
Now, we will form a two pair grouping of the multiples of 300, we get
$\Rightarrow 300=\underline{2\times 2}\times 3\times \underline{5\times 5}$
Now, taking the square root on both sides in the above equation, we get
$\Rightarrow \sqrt{300}=\left( 2\times 5 \right)\sqrt{3}$
$\Rightarrow \sqrt{300}=10\sqrt{3}$ ----------- (4)
Now, we will substitute the value of equation (2), (3), and (4) in expression (1), we get
$\Rightarrow 3\left( 3\sqrt{3} \right)+4\left( 2\sqrt{3} \right)-\left( 10\sqrt{3} \right)$
On further simplify the above expression, we get
$\Rightarrow 9\sqrt{3}+8\sqrt{3}-10\sqrt{3}$
Now, we see that $\sqrt{3}$ is common in all the terms of the above expression, we get
$\Rightarrow \sqrt{3}\left( 9+8-10 \right)$
On further simplification, we get
$\Rightarrow \sqrt{3}\left( 7 \right)$
Therefore, we get
$\Rightarrow 7\sqrt{3}$ which is the required solution.
Therefore, for the given expression $3\sqrt{27}+4\sqrt{12}-\sqrt{300}$ , its simplified value is equal to $7\sqrt{3}$.

Note:
While solving this problem, do mention the rules you are using to avoid confusion and mathematical errors. Always find the least common multiple of all the square roots and solve them step-by-step to get an accurate answer.