Question

What is the square root of $8$ in the radical form ?

Answer 1

Hint: We can see this problem is from indices and powers. In the given problem, we have to evaluate and simplify the square root of $8$ in the simplest radical form possible. We know that the square root of a number can be interpreted as the number being raised to the power $\left( {\dfrac{1}{2}} \right)$. So, we have, $8$ as the base and $\left( {\dfrac{1}{2}} \right)$ as the power. But we have to simplify this into radical form.

Complete step by step solution:
We have, $\sqrt 8 $
Expressing in simplest radical form is nothing but simplifying the radical into the simplest form with as few square roots, cube roots, etc as possible. left to find. In other words, a number under a radical is indivisible by a perfect square other than $1$.
So, we first factorize the number $8$ into its prime factors using the prime factorization method.
So, we get, $8 = 2 \times 2 \times 2$
Now, $\sqrt 8 = \sqrt {2 \times 2 \times 2} $
So, we can write $8$ as ${2^3}$.
Hence, we get, $\sqrt 8 = \sqrt {{2^3}} $
This is of the form ${a^{\left( {\dfrac{m}{n}} \right)}}$. But we can rewrite the expression using the laws of indices and powers ${a^{\left( {\dfrac{m}{n}} \right)}} = {\left( {{a^m}} \right)^{\dfrac{1}{n}}}$ .
Thus, we will apply same on the question above, we get,
$ \Rightarrow \sqrt 8 = \sqrt {{2^3}} = {2^{\dfrac{3}{2}}}$
Now, we can write ${2^{\dfrac{3}{2}}}$ as ${2^{\left( {1 + \dfrac{1}{2}} \right)}}$.
$ \Rightarrow \sqrt 8 = {2^{\left( {1 + \dfrac{1}{2}} \right)}}$
Also, we know the property ${a^{m + n}} = {a^m}{a^n}$. So, we get, ${2^{\left( {1 + \dfrac{1}{2}} \right)}} = {2^1} \times {2^{\dfrac{1}{2}}}$.
Now, we get,
$ \Rightarrow \sqrt 8 = \left( {{2^1}} \right)\left( {{2^{\dfrac{1}{2}}}} \right)$
Since we know that ${a^{\dfrac{1}{2}}} = \sqrt a $.
Hence, simplifying further, we get,
$ \Rightarrow \sqrt 8 = 2\sqrt 2 $
Therefore, the square root of $8$ in the radical form is $2\sqrt 2 $.

Note:
These rules or laws of indices help us to minimize the problems and get the answer in very less time. These powers can be positive and negative but can be molded according to our convenience while solving the problem. Also note that cube-root, square-root can be represented as powers with fractions having $1$ as numerator and respective root in denominator.