Question

How do you simplify $\sqrt{3{{x}^{3}}}$ ?

Answer 1

Hint: Here, we first try to find any presence of perfect squares. Not having found any, we break down the ${{x}^{3}}$ term into ${{x}^{2}}\times x$ where ${{x}^{2}}$ is a perfect square. We then simplify the remaining expression by taking $\sqrt{{}}$ as power $\dfrac{1}{2}$ and then evaluate to the final answer.

Complete step by step answer:
The given expression that we have at our disposal is,
$\sqrt{3{{x}^{3}}}$
Generally, the roots are written in the standard form $\sqrt[n]{{}}$ which indicates the ${{n}^{th}}$ root of the expression. Following this trend, the square root must be written as $\sqrt[2]{{}}$ , but instead it is written as simply $\sqrt{{}}$ . So, whenever only $\sqrt{{}}$ is written, we have to understand that it is square root. Square root can be any number of expressions which may or may not be a perfect square. For perfect squares, the square root of the expression or number is a bit simpler. So, if we are given to evaluate the square root of an expression, at first we need to check if it's a perfect square or not.
Our expression is $3{{x}^{3}}$ . We can see that $3$ is not a perfect square. Also, it being a prime number, it cannot be expressed as a product of two numbers. So, we leave it as it is. ${{x}^{3}}$ can be expressed as ${{x}^{2}}\times x$ . Now, ${{x}^{2}}$ is a perfect square and it is the square of $x$ . Thus, the given expression can be written as,
\[\begin{align}
  & \Rightarrow 3{{x}^{3}}=3\times {{x}^{2}}\times x \\
 & \Rightarrow \sqrt{3{{x}^{3}}}=\sqrt{3\times {{x}^{2}}\times x} \\
 & \Rightarrow \sqrt{3{{x}^{3}}}={{3}^{\dfrac{1}{2}}}\times {{x}^{2\times \dfrac{1}{2}}}\times {{x}^{\dfrac{1}{2}}} \\
 & \Rightarrow \sqrt{3{{x}^{3}}}=\sqrt{3}\times x\times \sqrt{x} \\
 & \Rightarrow \sqrt{3{{x}^{3}}}=\sqrt{3}{{x}^{\dfrac{3}{2}}} \\
\end{align}\]
Therefore, we can conclude that the expression can be simplified to \[\sqrt{3}{{x}^{\dfrac{3}{2}}}\] .

Note:
In this question, none of the terms are perfect squares, but we have somehow managed to express them in the closest perfect square by breaking down the algebraic term. We should always follow this technique for these types of problems. We can also express the final answer as $x\sqrt{3x}$ .