Question

What is the value of $\sqrt {16 + \sqrt {80 + \sqrt {5000 - 4999} } } $?

Answer 1

Hint: Here in this question, we need to find a value of given nested radicals. For this, first we need to start simplifying the radical from the inner side. To simplify, a radical should know the square numbers and apply arithmetic operations according to the ‘BODMAS’ rule then further simplification we get the required value.

Complete step by step answer:
The square root of a natural number is a value, which can be written in the form of $$y = \sqrt a $$. It means ‘y’ is equal to the square root of a, where ‘a’ is any natural number. We can also express it as $${y^2} = a$$. Thus, it is concluded here that square root is a value which when multiplied by itself gives the original number, i.e., $$a = y \times y$$.
The symbol or sign to represent a square root is ‘$$\sqrt {} $$’. This symbol is also called a radical. Also, the number under the root is called a radicand.
Consider the given radical
$$\sqrt {16 + \sqrt {80 + \sqrt {5000 - 4999} } } $$
We need to simplify from the inner side of radicals
Now, subtract 4999 from 5000. So, we get 1.
$$ \Rightarrow \,\,\,\,\sqrt {16 + \sqrt {80 + \sqrt 1 } } $$
As we know, the square root of 1 is 1 i.e., $$\sqrt 1 = 1$$.
$$ \Rightarrow \,\,\,\,\sqrt {16 + \sqrt {80 + 1} } $$
Add 80 and 1. So, we get 81.
$$ \Rightarrow \,\,\,\,\sqrt {16 + \sqrt {81} } $$
As we know, the square root of 81 is 9 i.e., $$\sqrt {81} = 9$$.
$$ \Rightarrow \,\,\,\,\sqrt {16 + 9} $$
Add 16 and 9. So, we get 25.
$$ \Rightarrow \,\,\,\,\sqrt {25} $$
As we know, the square root of 25 is 5 i.e., $$\sqrt {25} = 5$$.
$$ \Rightarrow \,\,\,\,5$$
$$\therefore \,\,\,\,\,\,\sqrt {16 + \sqrt {80 + \sqrt {5000 - 4999} } } = 5$$
Therefore, the value of $$\sqrt {16 + \sqrt {80 + \sqrt {5000 - 4999} } } $$ is $5$.

Note:
Remember, when simplifying the nested radicand, solve the radical one by one from the inner radical. And should know the square and square root numbers at least from 1 to 100. We can also simplify radicand by converting to the exponential numbers then by law of indices we can solve the given number.