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,
$
  {{\text{x}}^{\text{2}}}{\text{ = 72 + x}} \\
  {{\text{x}}^{\text{2}}}{\text{ - x - 72 = 0}} \\
 $
We get the value of x by solving the quadratic equation. We can solve the equation by factorization. We get,
$
  {{\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}} \\
 $
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.