Question

How do you simplify $\sqrt{125n}$?

Answer 1

Hint: We first explain the square root value and how the formulas of indices work for square root. We take the prime factorisation of the constant term 125 of $\sqrt{125n}$. We take the multiplication and use other formulas to find the root value.

Complete step by step answer:
We are going to simplify the given root value of $125n$. The term is multiple of a constant term 125 and a variable n.
We break the constant into its prime factorisation. We keep the variable as it is. We can’t break it as not enough information about n is given.
The prime factorisation of 125 gives
$\begin{align}
  & 5\left| \!{\underline {\,
  125 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & 1\left| \!{\underline {\,
  5 \,}} \right. \\
\end{align}$
This gives $125=5\times 5\times 5={{5}^{3}}$.
We know the formula of indices where $\sqrt[n]{x}={{\left( x \right)}^{\dfrac{1}{n}}}$. The ${{n}^{th}}$ root can be expressed in both ways.
If the value of n is not mentioned then we always assume that the value of n is 2. It always represents square root values.
Therefore, $\sqrt{125n}={{\left( 125n \right)}^{\dfrac{1}{2}}}$.
From the prime factorisation we can write $125n={{5}^{3}}n$.
We also have the formulas like ${{\left( xy \right)}^{a}}={{x}^{a}}{{y}^{a}};{{\left( {{x}^{m}} \right)}^{\dfrac{1}{n}}}={{x}^{\dfrac{m}{n}}}$.
We take square root on the both sides of the equation $125n={{5}^{3}}n$.
${{\left( 125n \right)}^{\dfrac{1}{2}}}={{\left( {{5}^{3}}n \right)}^{\dfrac{1}{2}}}$
Now applying the formula, we get ${{\left( {{5}^{3}}n \right)}^{\dfrac{1}{2}}}={{5}^{\dfrac{3}{2}}}{{n}^{\dfrac{1}{2}}}=5\times {{5}^{\dfrac{1}{2}}}{{n}^{\dfrac{1}{2}}}=5\sqrt{5n}$.
Therefore, \[\sqrt{125n}={{\left( 125n \right)}^{\dfrac{1}{2}}}=5\sqrt{5n}\].

The simplified value of $\sqrt{125n}$ is \[5\sqrt{5n}\].

Note: We can keep the root value of $\dfrac{1}{2}$ as their power value. The process of taking another variable x as the solution of $\sqrt{125n}$ where $x=\sqrt{125n}$. But if we take the square of the given equation then we are increasing the number of roots of the equation. Therefore, solving it directly is preferred.