Find the value of √ (13√(13√(13√(13√(13√(13√(13))))))).

Step-by-Step Solution

To solve an expression with nested square roots of the form: sqrt(x * sqrt(x * sqrt(x * ...))) (repeated n times), we use the following formula: x ^ ((2^n - 1) / 2^n)

1. Identify the variables:

   • The number being repeated (x) is 13.

   • Count the number of square roots (n).

   • Let's count them in the expression: sqrt(13 * sqrt(13 * sqrt(13 * sqrt(13 * sqrt(13 * sqrt(13 * sqrt(13)))))))

   • There are 7 nested radicals.

2. Apply the formula:

   • n = 7.

   • 2^n = 2^7 = 128.

   • The exponent is (2^7 - 1) / 2^7 = (128 - 1) / 128 = 127/128.

3. Final Value:

   • The expression evaluates to 13^(127/128).