Question

How do you simplify \[\sqrt {1125} \]?

Answer 1

Hint: In the given question, we have been asked to find the simplified form of the square root of an even natural number. To solve this question, we just need to know how to solve the square root. If the number is a perfect square, then it will have no integer left in the square root. But if it is not a perfect square, then it has at least one integer in the square root.

Complete step by step answer:
The given number whose simplified form is to be found is the square root of \[1125\], or we have to evaluate the value of \[\sqrt {1125} \].
First, we find the prime factorization of \[1125\] and club the pair(s) of equal integers together.
\[\begin{array}{l}3\left| \!{\underline {\,
  {1125} \,}} \right. \\3\left| \!{\underline {\,
  {375} \,}} \right. \\5\left| \!{\underline {\,
  {125} \,}} \right. \\5\left| \!{\underline {\,
  {25} \,}} \right. \\5\left| \!{\underline {\,
  5 \,}} \right. \\{\rm{ }}\left| \!{\underline {\,
  1 \,}} \right. \end{array}\]
Hence, \[1125 = 3 \times 3 \times 5 \times 5 \times 5 = {3^2} \times {5^2} \times 5 = 5 \times {15^2}\]
Hence, \[\sqrt {1125} = \sqrt {{{\left( {15} \right)}^2} \times 5} = 15\sqrt 5 \]

Thus, the simplified form of \[\sqrt {1125} \] is \[15\sqrt 5 \].

Note: In the given question, we had to find the value of the square root of a number. The questions of such type are pretty straight-forward. The methodology of solving is specific – factorize the number into prime factors, for each two equal numbers, club them as one and then multiply all the clubbed numbers. Students make mistakes while clubbing the numbers – would club more than two, or would make mistakes while multiplying the numbers. So, care must be taken at that point. It is not necessary that all the numbers are going to have perfect squares. If some numbers cannot be paired, they are to be left behind in the square root bracket.