![How To Evaluate √6√6√⋯ [duplicate]](https://netmathematik.de/image/how-to-evaluate-6-6-duplicate.jpeg "How To Evaluate √6√6√⋯ [duplicate]")
The expression √6√6√⋯, where the square root symbol continues infinitely, might seem daunting at first. However, by employing a bit of mathematical ingenuity, we can unravel this seemingly endless calculation and arrive at a definitive answer. The key lies in recognizing the recursive nature of this expression and utilizing the properties of square roots to solve it.
Understanding the Recursion
The expression √6√6√⋯ represents a never-ending sequence of nested square roots. Each square root takes the previous one as its input. This recursive nature is crucial to understanding how to evaluate the expression.
Solving for the Value
Let's denote the value of the entire expression as x. We can write the expression as:
x = √6√6√⋯
Notice that the expression inside the first square root is the same as the entire expression itself. Therefore, we can substitute x into the expression:
x = √6x
Now, we have a simple equation to solve for x. Squaring both sides of the equation:
x² = 6x
Rearranging the equation, we get a quadratic equation:
x² - 6x = 0
Factoring out an x:
x(x - 6) = 0
This gives us two possible solutions:
x = 0 or x = 6
However, since we are dealing with a series of square roots, the value of x cannot be 0. This is because the square root of 0 is 0, and the expression would become infinitely small and converge to 0, not 6.
Therefore, the only valid solution for x is:
x = 6
Verification
To verify our solution, we can plug 6 back into the original expression:
√6√6√⋯ = √6(√6√6√⋯) = √6(6) = √36 = 6
This confirms that our solution, x = 6, is indeed correct.
Conclusion
The seemingly infinite expression √6√6√⋯ has a finite value of 6. This is achieved by recognizing the recursive nature of the expression and using the properties of square roots to simplify the calculation. The process demonstrates how seemingly complex mathematical problems can be solved with clever techniques and a good understanding of fundamental concepts.
