Question

How do you simplify $\sqrt{4{{a}^{2}}}$ ?

Answer 1

Hint: At first, we express $4{{a}^{2}}$ as a perfect square, that is $4{{a}^{2}}={{\left( 2a \right)}^{2}}$ . Then, we transform the square root symbol into the index $\dfrac{1}{2}$ and then simplify the expression to $2a$ .

Complete answer:
The given expression that we have at our disposal is,
$\sqrt{4{{a}^{2}}}$
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 of 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 $4{{a}^{2}}$ . Now, we can see that $4$ is the square of the natural number $2$ and ${{a}^{2}}$ is nothing but the square of $a$ . This means that the entire expression $4{{a}^{2}}$ is nothing but the square of the expression $2\times a$ or simply $2a$ . This means that $4{{a}^{2}}$ is a perfect square and can be written as
$\Rightarrow 4{{a}^{2}}={{\left( 2a \right)}^{2}}$
Thus,
$\begin{align}
  & \Rightarrow \sqrt{4{{a}^{2}}}=\sqrt{{{\left( 2a \right)}^{2}}} \\
 & \Rightarrow \sqrt{4{{a}^{2}}}={{\left( 2a \right)}^{2\times \dfrac{1}{2}}} \\
 & \Rightarrow \sqrt{4{{a}^{2}}}=2a \\
\end{align}$
Therefore, we can conclude that, $\sqrt{4{{a}^{2}}}$ can be simplified to $2a$ .

Note:
At the beginning of simplification, we should always check if the expression under the root is a perfect square or not. This saves a lot of time and effort as in this case. For non perfect expressions, we terminate the expression with a power like ${{a}^{b}}$ .