Question

What is the square root of $90$ simplified in radical form?

Answer 1

Hint: The expression to be simplified is given in the above question as a mathematical statement, which is “square root of $90$”. In the question, it is given that we are supposed to simplify it in the radical form. Therefore, we first have to write the mathematical expression corresponding to the given statement in the radical form. So we will write the mathematical expression as $\sqrt{90}$. Then to simplify the mathematical expression, we have to consider the prime factorization of $90$, and write it as the multiplication of the prime factors. Then by using the radical law $\sqrt{ab}=\sqrt{a}\sqrt{b}$, we will be able to split the given expression as a multiplication of the radicals.

Complete step-by-step solution:
According to the question, we have to simplify the square root of $90$ in the form of radical. For this, we first write it in the radical form as
$\Rightarrow \sqrt{90}$
Now, to simplify it, we consider the prime factorization of $90$ using the long division method as shown below.
$\begin{align}
  & 2\left| \!{\underline {\,
  90 \,}} \right. \\
 & 5\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
From the above prime factorization, we can write $90=2\times 5\times 3\times 3$. Putting this in the given expression, we will obtain it as
\[\begin{align}
  & \Rightarrow \sqrt{2\times 5\times 3\times 3} \\
 & \Rightarrow \sqrt{10\times {{3}^{2}}} \\
\end{align}\]
Now, we can use the radical law given as $\sqrt{ab}=\sqrt{a}\sqrt{b}$ to write the above expression as
$\Rightarrow \sqrt{10}\sqrt{{{3}^{2}}}$
Now, since the square root and the square over a number cancel each other, we can write the above expression as
\[\begin{align}
  & \Rightarrow \sqrt{10}\times 3 \\
 & \Rightarrow 3\sqrt{10} \\
\end{align}\]
Hence, we have finally simplified the given mathematical expression as $3\sqrt{10}$.

Note: We must note that we have multiplied the unequal factors $2$ and $5$ inside the radical. This is because only the square over a number can cancel out the radical. The unequal factors thus have to be multiplied with each other inside the radical.