Question

How do you simplify $\sqrt{2x}.\sqrt{3x}$ ?

Answer 1

Hint: At first, we convert the $\sqrt{{}}$ to power $\dfrac{1}{2}$ . After that, we apply the two properties of indices which are ${{a}^{m}}{{b}^{m}}={{\left( ab \right)}^{m}}$ and ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ to the individual terms. After a little simplification, we get the desired result.

Complete step by step answer:
The given expression that we have at our disposal is,
$\sqrt{2x}.\sqrt{3x}$
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. Square root is equivalent to a power $\dfrac{1}{2}$ . Thus, the expression can be written as,
$\Rightarrow {{\left( 2x \right)}^{\dfrac{1}{2}}}.{{\left( 3x \right)}^{\dfrac{1}{2}}}$
Simplifying the above expression, we get,
$\Rightarrow \left( {{2}^{\dfrac{1}{2}}}{{x}^{\dfrac{1}{2}}} \right).\left( {{3}^{\dfrac{1}{2}}}{{x}^{\dfrac{1}{2}}} \right)$
Now, we know that an expression of the form ${{a}^{m}}{{b}^{m}}$ can be written as ${{\left( ab \right)}^{m}}$ . So, we can write ${{2}^{\dfrac{1}{2}}}\times {{3}^{\dfrac{1}{2}}}$ as ${{\left( 2\times 3 \right)}^{\dfrac{1}{2}}}$ which is equal to ${{6}^{\dfrac{1}{2}}}$ . The expression thus becomes
$\Rightarrow {{6}^{\dfrac{1}{2}}}\times {{x}^{\dfrac{1}{2}}}\times {{x}^{\dfrac{1}{2}}}$
Now, we know that an expression of the form ${{a}^{m}}\times {{a}^{n}}$ can be written as ${{a}^{m+n}}$ . So, we can write ${{x}^{\dfrac{1}{2}}}\times {{x}^{\dfrac{1}{2}}}$ can be written as ${{x}^{\dfrac{1}{2}+\dfrac{1}{2}}}$ which is equal to $x$ . Thus, the expression can be written as,
$\Rightarrow {{6}^{\dfrac{1}{2}}}\times x$
Replacing the power $\dfrac{1}{2}$ with $\sqrt{{}}$ , we get,
$\Rightarrow \sqrt{6}\times x$
That is equal to $x\sqrt{6}$ .
Therefore, we can conclude that the given expression can be simplified to $x\sqrt{6}$ .

Note:
In these types of problems, it is always better to at first convert the $\sqrt{{}}$ to power $\dfrac{1}{2}$ instead of operating within the brackets. This lowers the chances of errors. In the end, we should check whether the arithmetic term or the algebraic terms have a perfect square factor or not. If there, we should take it out of the root, like $\sqrt{8}=2\sqrt{2}$ .