Question

How do you simplify $\sqrt{49{{x}^{5}}}$ ?

Answer 1

Hint: At first, we look for the perfect square terms in the expression and then express them in the square form like $49$ as ${{7}^{2}}$ and so on. We then express ${{x}^{5}}$ as ${{x}^{4}}\times x$ in which ${{x}^{4}}$ is again a perfect square, after simplifying, we reach the final answer.

Complete step by step answer:
The given expression that we have at our disposal is,
$\sqrt{49{{x}^{5}}}$
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 or expression 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 $\sqrt{49{{x}^{5}}}$ . Now, we can see that $49$ is the square of the natural number $7$ . ${{x}^{5}}$ can be written as ${{x}^{4}}\times x$ . ${{x}^{4}}$ is nothing but the square of ${{x}^{2}}$ . This means that the entire expression $49{{x}^{5}}$ is nothing but the square of the expression $7\times {{x}^{2}}$ multiplied with $x$ and can be written as
$\Rightarrow 49{{x}^{5}}={{\left( 7{{x}^{2}} \right)}^{2}}x$
Thus,
$\begin{align}
  & \Rightarrow \sqrt{49{{x}^{5}}}=\sqrt{{{\left( 7{{x}^{2}} \right)}^{2}}x} \\
 & \Rightarrow \sqrt{49{{x}^{5}}}={{\left( 7{{x}^{2}} \right)}^{2\times \dfrac{1}{2}}}\times {{x}^{\dfrac{1}{2}}} \\
 & \Rightarrow \sqrt{49{{x}^{5}}}=\left( 7{{x}^{2}} \right)\times \sqrt{x} \\
 & \Rightarrow \sqrt{49{{x}^{5}}}=7{{x}^{2}}\sqrt{x} \\
\end{align}$
Therefore, we can conclude that the expression $\sqrt{49{{x}^{5}}}$ can be simplified to $7{{x}^{2}}\sqrt{x}$ .

Note:
We should look for the power of $x$ and see if it is a multiple of $2$ or not, such as in this case. The final simplified form can be written as either $7{{x}^{2}}\sqrt{x}$ or $7{{x}^{\dfrac{5}{2}}}$ as both are correct. Also, we should check if the arithmetic term is a perfect square or not.