Theory:

We tend to approximate some values to understand or use that value in our day-to-day lives easier.

Let us understand this concept through a real-life example.

The expenditure of the family for a month is $$12990$$.

The  family is approximately spending $$13000$$ per month.

What do you think about the above two statements? Both statements mean the same thing, right?

Sometimes, we use this approximate, rounded value for better understanding.

Like this, we can round off the values of the square root.

If the given number is not a perfect square, we estimate the approximate square root value for that number.
Example:
Find the approximate value of $$\sqrt{78}$$.

The closest perfect squares near $$78$$ are $$64$$ and $$81$$.

$$\sqrt{64} = 8$$ and $$\sqrt{81} = 9$$

The number $$78$$ lies between the square of $$64$$ and $$81$$.

$$\Rightarrow 64 < 78 < 81$$

$$\Rightarrow \sqrt{64} < \sqrt{78} < \sqrt{81}$$

$$\Rightarrow \sqrt{8^2} < \sqrt{78} < \sqrt{9^2}$$

$$\Rightarrow 8 < \sqrt{78} < 9$$    (Square and square root get cancelled)

In comparison, $$78$$ stays near to $$81$$, which is the square of $$9$$.

Therefore, we can write the approximate square root of $$78$$ as $$9$$.