Question

What is the square root of $69?$

Answer 1

Hint: We will check if the given number is a perfect square. If it is a perfect square, we can directly find a rational number which is the square root of the given number. If the given number is not a perfect square, we need to check if it is a multiple of perfect square. If it is, then we can simplify the given number to a rational number which is the square root of the given number.

Complete step by step solution:
Let us consider the given number, $69.$
We are asked to find the square root of this number.
We know that the given number is not a perfect square. Let us check if we can write the given number as a multiple of a perfect square in order to simplify the given number to a further simplified radical form.
Since $69$ is not a prime number, we know that there exist factors that are not $1$ and $69.$
Let us find the factors of $69.$
We will get $1,3,23$ and $69.$
We can see that apart from $1,$ there is no other perfect square among the factors of $69.$
So, it is clear that the square root of the given number cannot be simplified.
Hence $\sqrt{69}$ is in its simplest radical form.

Note: The square root of $69$ can be calculated as a decimal number. So, we will get the square root of $69$ is approximately equal to $8.307.$ If the given number is a multiple of a perfect square, we can use the identity $\sqrt{ab}=\sqrt{a}\sqrt{b}.$