Question

How do you simplify \[\sqrt {76} \] ?

Answer 1

Hint: In the given question, we have been asked to find the simplified form of an even natural number, inside a square root. 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 \[\sqrt {76} \].
First, we find the prime factorization of \[76\] and club the pair(s) of equal integers together.
\[\begin{array}{l}{\rm{ }}2\left| \!{\underline {\,
  {76} \,}} \right. \\{\rm{ }}2\left| \!{\underline {\,
  {38} \,}} \right. \\19\left| \!{\underline {\,
  {19} \,}} \right. \\{\rm{ }}\left| \!{\underline {\,
  1 \,}} \right. \end{array}\]
Hence, \[76 = 2 \times 2 \times 19 = {2^2} \times 19\]
Hence, \[\sqrt {76} = \sqrt {{{\left( 2 \right)}^2} \times 19} = 2\sqrt {19} \]
Thus, the simplified form of \[\sqrt {76} \] is \[2\sqrt {19} \].

Note: When we are calculating such questions, we find the prime factorization, club the pairs together, take them out as a single number and solve for it. This procedure requires no further action or steps to evaluate the answer. It is a point to remember that a perfect square always has an even number of factors.