Question

What is the square root of 150 in simplified radical form?

Answer 1

Hint: We are asked to write the square root of 150 in simplified radical form and which we are going to calculate by writing the prime factorization of 150 and then we will arrange the factorization in such a manner so that we will get some even exponents of the factors. And then we can put those factors outside the square root symbol and in this way, we will get the simplified radical form.

Complete step-by-step solution:
To write the square root of 150 in the simplified radical form, we are going to first of all write 150 in square root form as follows:
$\sqrt{150}$
After that, we will do the prime factorization of the number 150. The prime factorization of 150 is as follows:
$150=2\times 3\times 5\times 5$
Substituting the above factor form in place of 150 in $\sqrt{150}$ and we get,
$\sqrt{2\times 3\times 5\times 5}$
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:
$\sqrt{2\times 3\times 5^2}$
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:
${(2\times 3\times 5^2)}^{{\displaystyle \frac{1}{2}}}$
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,
$2^{{\displaystyle \frac{1}{2}}}\times 3^{{\displaystyle \frac{1}{2}}}\times 5^{2\times {\displaystyle \frac{1}{2}}}$
The 2 written in the numerator and the denominator in the exponent of 5 will be cancelled out and we get,
$\begin{array}{rl}& 2^{{\displaystyle \frac{1}{2}}}\times 3^{{\displaystyle \frac{1}{2}}}\times 5\\ & =5\sqrt{2\times 3}\\ & =5\sqrt{6}\end{array}$
Hence, the square root of the 150 in simplified radical form is $5\sqrt{6}$.

Note: The simplified radical form which we have solved above is correct or not which we can check by taking the square of the calculated simplified radical form and see if we are getting the same number which is given in the above problem.
The square root of 150 in simplified radical form which we have calculated above is $5\sqrt{6}$. Now, multiplying $5\sqrt{6}$ by itself we get,
$5\sqrt{6}\times 5\sqrt{6}$
We know that when two square root will get multiplied then we get the original number so multiplying $\sqrt{6}$ by itself we get 6 so applying this understanding in the above we get,
$25\times 6=150$