Question

Evaluate the square root of \[128\] in radical form.

Answer 1

Hint: The basic and primary step to be thought in questions of square root is that we should remember all the squares of \[1\] to \[20\] .With practice \[20\] can be reminded but till \[10\] it is very helpful because if we consider this question in which \[96\] is given. So, we check numbers nearby \[96\] that is \[100\] .Now clearly \[96\] is less than \[100\] so square root of \[96\] is less than \[10\] and \[96\] is greater than \[81\] which means it’s square root is larger than \[9\] .So by doing this we can get a rough idea about the correct answer. As \[96\] is not a perfect square . Therefore, by prime factorization of \[96\] we will get \[5\] times \[2\] and one time \[3\] . Then after pairing, the number \[4\] comes out of the square root and \[6\] remains inside the square root. In the end we will get \[4\sqrt 6 \] as a square root of \[96\] in a radical form.

Complete step-by-step answer:
 Square root of \[96\] is not an integer so it is not a perfect square. Square root of the number \[96\] is a non-repeating and non-terminating number. So, \[\sqrt {96} \] is an irrational number.
Now, we will find the square root of \[96\] by using the prime factorization method.
For this divide the number \[96\] by the smallest prime factor that is \[2\] .
 \[\dfrac{{96}}{2} = 48\]
Again dividing \[48\] by \[2\]we get
 \[\dfrac{{48}}{2} = 24\]
Again dividing \[24\] by \[2\] we get
\[\dfrac{{24}}{2} = 12\]
Again dividing \[12\] by \[2\] we get
\[\dfrac{{12}}{2} = 6\]
Again dividing \[6\] by \[2\] we get
\[\dfrac{6}{2} = 3\]
Again dividing \[3\] by \[3\] we get
\[\dfrac{3}{3} = 1\]
On combining the factors we have \[\sqrt {96} \] \[ = \] \[\sqrt {2 \times 2 \times 2 \times 2 \times 2 \times 3} \]
On pairing the factors under the square root we get,
\[\sqrt {96} \] \[ = \] \[\sqrt {\left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times 2 \times 3} \]
\[ \Rightarrow 2 \times 2\sqrt {2 \times 3} \]
\[ \Rightarrow 4\sqrt 6 \]
Hence the square root of \[\sqrt {96} \] in radical form is \[4\sqrt 6 \] .
So, the correct answer is “ \[4\sqrt 6 \]”.

Note: The square root of a number can be positive or negative, real or imaginary. The square root of \[96\] is an irrational number. The square root of \[96\] lies between \[9\] and \[10\] . The square root of \[96\] is \[9.796\]. There are various methods to find the square root of a number like long division method, prime factorization method ..etc.