Question

What is $\sqrt{65}$ in simplified radical form?

Answer 1

Hint: To convert any expression into simplified radical form, we must know about the terminology used in this regard. We must find the prime factorisation of the number inside the radical sign, and then look for the prime numbers that are appearing at least twice. We can bring such prime numbers outside the radical sign to get the simplified radical form.

Complete step by step solution:
We know that a radical is nothing but the mathematical opposite of exponents. Square root is the smallest radical. Any expression that contains a root, is called a radical expression. The expression or number inside the root symbol is called a radicand. We must also note that the small number written in front of the root symbol is called the index of the radical.
A simplified radical form is a method of representation of radicals in which the number inside the radical symbol does not contain any square, or cube of any number, as per the index of that radical.
For square roots, we have index = 2.
Thus, we now know that to be in simplified radical form, the number inside the radical sign must not contain a square of any number.
In this problem, we have to express $\sqrt{65}$ in simplified radical form.
First, let us find the prime factorial of 65.
$\begin{align}
  & \text{ }5\left| \!{\underline {\,
  65 \,}} \right. \\
 & 13\left| \!{\underline {\,
  13 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Hence, we have $65=5\times 13$ .
In the prime factorisation, each prime number is occurring just once.
So, from this prime factorisation, we can see that 65 has no squares within.
Thus, we can clearly say that $\sqrt{65}$ is already in its simplified radical form.
Or, we can say that the simplified radical form of $\sqrt{65}$ is $\sqrt{65}$ .

Note: We must be very clear with the terminology used in context of the radicals. We must keep in mind that the prime factorisation of each number is unique. So, it is totally fine to comment about the number by analysing its prime factorial.