Question

How do you simplify the square root of $\dfrac{1}{8}$?

Answer 1

Hint: Assume the required expression as ‘E’. First of all we will write 8 as the product of its prime factors by using the prime factorization process. In the next step we will try to form groups of two similar prime factors and use the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to simplify. The factors which will not be grouped will be left as it is inside the radical sign. Finally, rationalize the denominator if there is any irrational number present there.

Complete step by step solution:
Here, we have been asked to simplify the square root of $\dfrac{1}{8}$.
Let us assume this expression as E. So, we have,
\[\Rightarrow E=\sqrt{\dfrac{1}{8}}\]
Since, we need to find the square root of $\dfrac{1}{8}$ so we will apply the prime factorization method to do this. In this method first we will write 8 as the product of its primes and try to form groups of two similar factors that will appear. So, we have,
$\Rightarrow 8=2\times 2\times 2$
Forming the group of two similar prime factors and writing them in the exponential form with exponent equal to 2, we get,
\[\Rightarrow 8={{2}^{2}}\times 2\]
Here, we are not able to group one of the factors (2) so it will be left as it is. Therefore, the expression ‘E’ becomes:
\[\begin{align}
  & \Rightarrow E=\sqrt{\dfrac{1}{{{2}^{2}}\times 2}} \\
 & \Rightarrow E={{\left( \dfrac{1}{{{2}^{2}}\times 2} \right)}^{\dfrac{1}{2}}} \\
\end{align}\]
Using the property of exponents given as ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$, we get,
\[\begin{align}
  & \Rightarrow E=\left( \dfrac{1}{{{2}^{2\times \dfrac{1}{2}}}\times {{2}^{\dfrac{1}{2}}}} \right) \\
 & \Rightarrow E=\left( \dfrac{1}{2\times {{2}^{\dfrac{1}{2}}}} \right) \\
 & \Rightarrow E=\left( \dfrac{1}{2\times \sqrt{2}} \right) \\
 & \Rightarrow E=\dfrac{1}{2\sqrt{2}} \\
\end{align}\]
Now, we need to rationalize the denominator because there is an irrational number present there. So, multiplying the numerator and the denominator with $\sqrt{2}$ we get,
\[\begin{align}
  & \Rightarrow E=\dfrac{1}{2\sqrt{2}}\times \dfrac{\sqrt{2}}{\sqrt{2}} \\
 & \Rightarrow E=\dfrac{\sqrt{2}}{2\times 2} \\
 & \Rightarrow E=\dfrac{\sqrt{2}}{4} \\
\end{align}\]
Hence, the above expression represents the simplified form of \[\sqrt{\dfrac{1}{8}}\].

Note: If you want you can simplify the expression \[\dfrac{\sqrt{2}}{4}\] a step further by substituting $\sqrt{2}=1.414$, but here we are not asked to do so therefore it does not matter here. The value of $\sqrt{2}$ is found using the long division method. You must remember the values of $\sqrt{2}$ and $\sqrt{3}$ as they are used in trigonometry. Remember the basic properties of exponents to make the calculations easy.