Question

How do you simplify \[\sqrt {125{x^2}} \] ?

Answer 1

Hint: When a number is multiplied itself, it results in the exponential form of the number as \[x \times x = {x^2}\]. Square root has a power of $\dfrac{1}{2}$ . If we are able to find the given numbers as squares of some other numbers, we can group them together and the powers of square and square root will cancel out each other.

Complete step-by-step solution:
If a number is multiplied by itself twice, the square root can be removed, i.e.
Square roots are evenly distributed over all the terms under it.
The given question is-
\[\sqrt {125{x^2}} \]
We will have to try to split and then group the numbers in such a way that we get these numbers in squared form. We wish to find the squared form because then we can use the exponential form to cancel out the square root using the following property-
$\sqrt {{x^2}} = {\left( x \right)^{2 \times \dfrac{1}{2}}} = {x^1} = x$
We can write $125$ as $5 \times 5 \times 5$ and we can write ${x^2}$ as $x \times x$ .
Therefore, the question can be written and the factors can be grouped as following-
\[ \Rightarrow \sqrt {\left( {\underbrace {5.5}_5} \right).5\left( {\underbrace {x.x}_x} \right)} \]
It is visible that we could group some terms and hence the square root was canceled but one 5 could not be grouped as there was a lack of another \[5\] . Therefore, this \[5\] will be retained in the square root. And the final answer is
\[ \Rightarrow 5x\sqrt 5 \]

Therefore, \[\sqrt {125{x^2}} \] = \[5\sqrt 5 \,x\]

Note: When a number is multiplied by itself, it results in the exponential form of the number as \[x \times x = {x^2}\], here, \[x\] is called the base and \[2\] is called the power or exponent. This form is read as \[x\] raised to the power two or \[x\] square. \[\sqrt {} \] , this symbol refers to the square root of the number under it. In simple words, one can understand it as if a number is under it in its squared form or with a power \[2\] , the value returned by the square root will be the number with its power as \[1\] . For example,
\[\sqrt {{x^2}} = x\], using numbers we can understand it as \[\sqrt 4 = 2\] . We can understand this as, \[4 = 2 \times 2\] ,so, \[\sqrt 4 \] can be written as $\sqrt {2 \times 2} $ which will return the value \[2\] .