Question

The value of $\sqrt{214+\sqrt{130-\sqrt{88-\sqrt{44+\sqrt{25}}}}}$
(a)14
(b)15
(c)16
(d)17

Answer 1

Hint: To solve the question given above, we will assume some value of all the 5 under roots. First we will calculate the value of innermost under root. Then we will calculate the second innermost under root, then the third innermost under root and so on.

Complete step-by-step answer:
To start with, we will assume that the value of $\sqrt{214+\sqrt{130-\sqrt{88-\sqrt{44+\sqrt{25}}}}}$ is ‘a’, the value of $\sqrt{130+\sqrt{88-\sqrt{44+\sqrt{25}}}}$ is ‘b’, the value of $\sqrt{88-\sqrt{44+\sqrt{25}}}$ is ‘c’, the value of $\sqrt{44+\sqrt{25}}$ is ‘d’ and the value of $\sqrt{25}$ is ‘e’. Thus we will get:
$a=\sqrt{214+\sqrt{130-\sqrt{88-\sqrt{44+\sqrt{25}}}}}..........\left( 1 \right)$
$b=\sqrt{130-\sqrt{88-\sqrt{44+\sqrt{25}}}}.......\left( 2 \right)$
$c=\sqrt{88-\sqrt{44+\sqrt{25}}}.........\left( 3 \right)$
$d=\sqrt{44+\sqrt{25}}.........\left( 4 \right)$
$e=\sqrt{25}........\left( 5 \right)$
Now, first we will calculate the value of ‘e’. We know that $\sqrt{25}=5$. Thus, we will get:
$e=5.......\left( 6 \right)$
Now, we know that, from (4) and (5) we have:
$d=\sqrt{44+e}.......\left( 7 \right)$
$\Rightarrow d=\sqrt{44+5}$
$\Rightarrow d=\sqrt{49}$
We know that $\sqrt{49}=7$. So, we will get:
$\Rightarrow d=7$
$\Rightarrow \sqrt{44+\sqrt{25}}=7.......\left( 8 \right)$
From (3) and (8), we will get the following equation:
$\Rightarrow c=\sqrt{88-7}$
$\Rightarrow c=\sqrt{81}$
We know that $\sqrt{81}=9$. Thus, we will get:
$\Rightarrow c=9$
$\Rightarrow \sqrt{88-\sqrt{44+\sqrt{25}}}=9.......\left( 9 \right)$
From (9) and (2), we will get the following equation:
$\Rightarrow b=\sqrt{130-9}$
$\Rightarrow b=\sqrt{121}$
Now, we know that $\sqrt{121}=11$. So we will get:
$\Rightarrow b=11$
$\Rightarrow \sqrt{130-\sqrt{88-\sqrt{44+\sqrt{25}}}}=11.........\left( 10 \right)$
From (10) and (1), we will get the following equation:
$\Rightarrow a=\sqrt{214+11}$
$\Rightarrow a=\sqrt{225}$
Now, we know that $\sqrt{225}=15$. Thus, we will get the following equation:
$\Rightarrow a=15$
$\Rightarrow \sqrt{214+\sqrt{130-\sqrt{88-\sqrt{44+\sqrt{25}}}}}=15$
Hence, the correct option is (b).

Note: If we do not know the values of square roots of the number given in question, we can calculate it by the method of prime factorisation. For example: The value of $\sqrt{215}$ is calculated by:

3	215
3	75
5	25
5	5
	1

$\Rightarrow 215=3\times 3\times 5\times 5$
$\Rightarrow 215=\left( 3\times 3 \right)\times \left( 5\times 5 \right)$
$\Rightarrow \sqrt{215}=3\times 5$
$\Rightarrow \sqrt{215}=15$
Similarly the values of other square roots can be calculated but the condition to apply this method is that the number inside the square root should be the square of a positive integer.