Question

What is the square root of 2500?

Answer 1

Hint: To find the square root of 2500 we are going to put $\dfrac{1}{2}$ in the exponent of 2500. Then we are going to find the prime factorization for 2500. And then we will convert the factorization in such a way so that we will get even powers in some or all the factors. Then the even power of the factors will get divided by 2 and in this way, we will get the square root of 2500.

Complete step-by-step solution:
In the above problem, we are asked to find the square root of 2500.
For that we are going to put square root in 2500 and we get,
$\sqrt{2500}$
Now, we are going to find the prime factorization for 2500. The prime factorization for 2500 is as follows:
$2500=5\times 5\times 5\times 5\times 2\times 2$
In the above equation, we can add the powers of the same bases which are written in the multiplication form so 5 is the same base and are written with the multiplication sign so their powers will get added and when 2 is the same base then we can add the power of 2 and we get,
$\begin{align}
  & 2500={{5}^{1+1+1+1}}\times {{2}^{1+1}} \\
 & 2500={{5}^{4}}\times {{2}^{2}} \\
\end{align}$
Now, substituting the above value of 2500 in the square root of 2500 we get,
$\sqrt{{{5}^{4}}\times {{2}^{2}}}$
We know that we can write $\dfrac{1}{2}$ to the power of something written inside the square root in place of square root symbol then the above expression will look as:
${{\left( {{5}^{4}}\times {{2}^{2}} \right)}^{\dfrac{1}{2}}}$
We also know the property of the exponents which says that:
${{\left( {{a}^{b}} \right)}^{n}}={{a}^{b\times n}}$
Applying the above property of exponents in the above expression we get,
$\begin{align}
  & \Rightarrow {{\left( {{5}^{4}}\times {{2}^{2}} \right)}^{\dfrac{1}{2}}} \\
 & ={{5}^{4\times \dfrac{1}{2}}}\times {{2}^{2\times \dfrac{1}{2}}} \\
\end{align}$
Now, the numerator and denominator of the powers of 5 and 2 will be divided by 2 and we get,
$={{5}^{2}}\times {{2}^{1}}$
Solving the above multiplication we get,
$25\times 2=50$
Hence, the square root of 2500 is equal to 50.

Note: We can check the square root of 2500 which we have calculated in the above solution by multiplying the square root with itself and then see whether we are getting 2500 or not.
The square root of 2500 which we have solved above is 50 so multiplying 50 by 50 we get,
$50\times 50=2500$
Hence, we are getting the same number which we have started with so the square root which we have solved above is correct.