Question

What is the square root of 164 simplified in radical form?

Answer 1

Hint: The square root of 164 simplified in radical form is calculated by writing the prime factorization of 164 and then we will arrange the factorization in such a fashion so that we will get some even exponents of the factors. And then we can put those factors outside the square root symbol and hence will get the simplified radical form.

Complete step-by-step solution:
In the above question, we are asked to write the square root of 164 in the simplified radical form so we are going to first of all write 164 in square root form as follows:
$\sqrt{164}$
After that, we will prime factorize the number 164. The prime factorization of 164 is as follows:
$164=2\times 2\times 41$
Now, substituting the above factor form in place of 164 in $\sqrt{164}$ and we get,
$\sqrt{2\times 2\times 41}$
We know that, when base is same and the base is written in the multiplication form then exponents will add up of the same base then the above square root expression will look like:
$\sqrt{2^2\times 41}$
We know that we can replace the square root symbol by the expression written inside the square root to the power of ${\displaystyle \frac{1}{2}}$. Then the above expression will look as follows:
${(2^2\times 41)}^{{\displaystyle \frac{1}{2}}}$
We also know that there is an exponent form which says that:
${\left(a^m\right)}^n=a^{m\times n}$
Applying the above exponent relation in the above multiplication expression we get,
$2^{2\times {\displaystyle \frac{1}{2}}}\times {41}^{{\displaystyle \frac{1}{2}}}$
The 2 written in the numerator and the denominator in the exponent of 2 will be cancelled out and we get,
$\begin{array}{rl}& 2^1\times {41}^{{\displaystyle \frac{1}{2}}}\\ & =2\sqrt{41}\end{array}$
Hence, the square root of the 164 in simplified radical form is $2\sqrt{41}$.

Note: We can check whether the simplified radical form for the square root of 164 is correct or not by taking the square of the calculated simplified radical form and see if we are getting the same number which we have started with.
The square root of 164 in simplified radical form which we have calculated above is $2\sqrt{41}$. Now, multiplying $2\sqrt{41}$ by itself we get,
$2\sqrt{41}\times 2\sqrt{41}$
We know that when two square root will get multiplied then we get the original number so multiplying $\sqrt{41}$ by itself we get 41 so applying this understanding in the above we get,
$4\times 41=164$