Question

What is the simplest radical form of 306?

Answer 1

Hint: To find the simplest radical form of 306 we will take its square root. Now, we will write 306 as the 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 306. That means we need to find the simplest radical form of the square root of 306. Let us assume the expression as E, so we have the radical expression,
$\Rightarrow E=\sqrt{306}$
Now, to simplify the above expression further we need to write 306 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,
\[\Rightarrow 306=2\times 3\times 3\times 17\]
Factor 3 is appearing twice so in exponential form we can write it as: -
\[\Rightarrow 306=2\times {{3}^{2}}\times 17\]
Therefore the expression E becomes: -
\[\begin{align}
  & \Rightarrow E=\sqrt{2\times {{3}^{2}}\times 17} \\
 & \Rightarrow E=\sqrt{2\times 17}\times \sqrt{{{3}^{2}}} \\
 & \Rightarrow E=\sqrt{34}\times {{\left( {{3}^{2}} \right)}^{\dfrac{1}{2}}} \\
\end{align}\]
Using the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ we get,
\[\begin{align}
  & \Rightarrow E=\sqrt{34}\times {{3}^{2\times \dfrac{1}{2}}} \\
 & \Rightarrow E=\sqrt{34}\times 3 \\
 & \therefore E=3\sqrt{34} \\
\end{align}\]
Hence, $3\sqrt{34}$ is the simplified radical form of 306.

Note: Do not try to find the value of $\sqrt{34}$ in the decimal form and substitute 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 all the basic formulas of ‘exponents and powers’ as they are used to simplify the radical expressions.