Question

What is the square root of 50 in simplified radical form?

Answer 1

Hint: To find the simplified radical form of square root of 50, first we will write it as a product of its prime factors. Now, if there will be the same prime factors more than once then we will write it in the exponential form and use the property of exponents ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to simplify the radical expression. The prime factors which may not be paired will be left inside the radical sign.

Complete step by step solution:
Here we have been asked to write the simplest radical form of square root of 50. Let us assume this expression as E. So we have the radical expression,
$\Rightarrow E=\sqrt{50}$
Now to simplify the above expression further we need to write it as the product of its prime factors. Since we have the under root sign in the radical expression so we will try to form pairs of two similar prime factors if they will appear and we will use the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to remove them from the radical sign. The factors which will appear only once will be left inside the radical sign. So we can write,
\[\Rightarrow 50=2\times 5\times 5\]
Factor 5 is appearing twice so in exponential form we can write it as:
\[\Rightarrow 50=2\times {{5}^{2}}\]
Therefore the expression E becomes:
\[\begin{align}
  & \Rightarrow E=\sqrt{2\times {{5}^{2}}} \\
 & \Rightarrow E=\sqrt{2}\times \sqrt{{{5}^{2}}} \\
 & \Rightarrow E=\sqrt{2}\times {{\left( {{5}^{2}} \right)}^{\dfrac{1}{2}}} \\
\end{align}\]
Using the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ we get,
\[\begin{align}
  & \Rightarrow E=\sqrt{2}\times {{5}^{2\times \dfrac{1}{2}}} \\
 & \Rightarrow E=\sqrt{2}\times 5 \\
 & \therefore E=5\sqrt{2} \\
\end{align}\]

Hence, $5\sqrt{2}$ is the simplified radical form of $\sqrt{50}$.

Note: Do not substitute the value of $\sqrt{2}=1.414$ in the obtained expression to find the product because we haven’t been asked to find the square root but we just have to simplify the expression. You must remember the formulas of exponents as they are used to simplify the radical expressions.