Question

The value of $\sqrt{\text{72 + }\sqrt{\text{72 + }\sqrt{\text{72 + }......}}}$ is:

Answer 1

Hint: We can equate the infinite nested root to a variable. The term inside the radical can be replaced with the variable. By taking the square and solving the variable, we get the required solution.

Complete step by step answer:

Here we have an infinite number of nested square roots. We can equate the whole thing to x.
$\text{x = }\sqrt{\text{72 + }\sqrt{\text{72 + }\sqrt{\text{72 + }......}}}$
As it is never ending, the term inside the 1st square root can be written as 72+x.
$\text{x = }\sqrt{\text{72 + x}}$
Now take the square root on both sides. We get a quadratic equation as follows,
$$\begin{gathered} {\text{x}}^{\text{2}}\text{ = 72 + x} \\ {\text{x}}^{\text{2}}\text{ - x - 72 = 0} \end{gathered}$$
We get the value of x by solving the quadratic equation. We can solve the equation by factorization. We get,
$$\begin{gathered} {\text{x}}^{\text{2}}\text{ - x - 72 = 0} \\ {\text{x}}^{\text{2}}\text{ - 9x + 8x - 72 = 0} \\ \text{x}\left(\text{x - 9}\right)\text{ + 8}\left(\text{x - 9}\right)\text{ = 0} \\ \left(\text{x - 9}\right)\left(\text{x + 8}\right)\text{ = 0} \end{gathered}$$
Therefore, x can have 2 values. $\text{x = 9, - 8}$
But, according to the question, we have x as the square root of some numbers. As square roots cannot be negative, x also cannot be negative. So, we take the value $\text{x = 9}$
So, the required solution is 9.

Note: This type of expression is known as infinite nested radicals. We can use this method for solving any infinite nested radical. Even though this is an infinite series, it has a finite sum. We must keep in mind that a number inside the square root radical will be always positive. If the number inside the square root radical is negative, then its value will become a complex number. We can solve the quadratic equation formed using any method as the solution will be unique.