Question

What is the square root of \[8100\]?

Answer 1

Hint: We are given to find the square root of \[8100\]. Firstly, we check whether the given number is a perfect square or not. We can find it by either long division or by prime factorization method. Square root exists only when a number is a perfect square. The square root can be either positive or negative because the product of two negative numbers is also a positive number.

Complete step by step answer:
Let us learn about square roots now. The square root is nothing but the number when squared or multiplied itself gives the original number. When a number is a perfect square, the perfect square root exists for that number. The square root of any negative number is undefined.
Now let us find out the square root of \[8100\].
Since it ends with zeroes, it is a perfect square.
Next, we are supposed to prime factorize the given number \[8100\]. Upon factoring we get,
\[\begin{align}
  & 5\left| \!{\underline {\,
  8100 \,}} \right. \\
 & 5\left| \!{\underline {\,
  1620 \,}} \right. \\
 & 3\left| \!{\underline {\,
  324 \,}} \right. \\
 & 3\left| \!{\underline {\,
  108 \,}} \right. \\
 & 3\left| \!{\underline {\,
  36 \,}} \right. \\
 & 3\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & \left| \!{\underline {\,
  2 \,}} \right. \\
\end{align}\]
Now this can be shown as product of primes i.e. \[8100={{5}^{2}}\times {{3}^{4}}\times {{2}^{2}}\]
Since we are given to find the square root, we will be expressing it in the following way.
Apply square root on both the sides, we get
\[\sqrt{8100}=\sqrt{{{5}^{2}}\times {{3}^{4}}\times {{2}^{2}}}\]
Upon solving this equation,
\[\begin{align}
  & \Rightarrow \sqrt{8100}=\sqrt{{{5}^{2}}\times {{3}^{2}}\times {{3}^{2}}\times {{2}^{2}}} \\
 & =5\times 3\times 3\times 2 \\
 & =90 \\
\end{align}\]
\[\therefore \] The square root of \[8100\] is \[90\].

Note: All non-zero numbers have only one real root. This method of prime factorization breaks down the number into sub parts which makes it easier for finding the root of the number. A perfect square cannot be negative. Here, since the last two digits are 00, we can guess that it will be a possibility that the required number is a multiple of 10 since the square of 10 is 100. Then, we have 81, which is the square of 9. So, we can say that the square root of 8100 would be 90.