Question

What is the square root of $180$ simplified in radical form?

Answer 1

Hint: For simplifying the square root of the given number in the radical form, we first have to write it in the form of radical. The given expression in the radical form will be written as $\sqrt{180}$. To simplify it, we need to consider the prime factorization of the number $180$. Then using the radical law given by $\sqrt{ab}=\sqrt{a}\sqrt{b}$, we will be able to obtain the expression as a product of radicals. Then finally on cancelling the squares with the square roots, we will obtain the expression simplified in the form of radical.

Complete step-by-step solution:
According to the question, we have to simplify the square root of $180$. For this, we first write it in the form of radical as
$\Rightarrow E=\sqrt{180}$
Now, in order to simplify it, we consider the prime factorization of the number $180$, as shown below.
\[\begin{align}
  & 2\left| \!{\underline {\,
  180 \,}} \right. \\
 & 2\left| \!{\underline {\,
  90 \,}} \right. \\
 & 3\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
From the above prime factorization, we can write the number $180$ as
\[\begin{align}
  & \Rightarrow 180=2\times 2\times 3\times 3\times 5 \\
 & \Rightarrow 180={{2}^{2}}\times {{3}^{2}}\times 5 \\
\end{align}\]
On substituting this in the given expression, we get
\[\Rightarrow E=\sqrt{{{2}^{2}}\times {{3}^{2}}\times 5}\]
Using the rule of radicals given by $\sqrt{ab}=\sqrt{a}\sqrt{b}$ we can simply the above expression as
\[\Rightarrow E=\sqrt{{{2}^{2}}}\times \sqrt{{{3}^{2}}}\times \sqrt{5}\]
We know that the squares and the square root cancel each other. So the above expression becomes
$\begin{align}
  & \Rightarrow E=2\times 3\times \sqrt{5} \\
 & \Rightarrow E=6\sqrt{5} \\
\end{align}$
Hence, the given expression has been finally simplified in the radical form as $6\sqrt{5}$.

Note: We may think of using the long division method to obtain the square root of five to get the final answer in the form of decimal. But we must note that the question has asked us to simplify the expression in the radical form, and not in the decimal form.