What is the Value of √3√3√3√3√3...

Exploring the fascinating world of infinite nested radicals can be a thrilling mathematical adventure. Today, we will delve into the intriguing question: What is the value of the expression #8730;3#8730;3#8730;3#8730;3... (read as √3√3√3√3√3...)? This is an excellent example of how simple mathematical concepts can lead to deep and profound results.

Solving for Infinite Nested Radicals

Let's denote the expression by x, such that x #8730;3#8730;3#8730;3.... We can express this in a recursive manner:

x sqrt{3 cdot sqrt{3 cdot sqrt{3 ...}}}

By the nature of the nested radical, we have:

x sqrt{3x} implies x^2 3x implies x^2 - 3x 0 implies x(x - 3) 0end{equation>

Solving for (x), we get two possible solutions:

x frac{1 pm sqrt{1 12}}{2} implies x frac{1 pm sqrt{13}}{2} end{equation>

However, since (x) is a square root and cannot be negative, we discard the negative solution, and we are left with:

x frac{1 sqrt{13}}{2} end{equation>

Therefore, the value of the expression #8730;3#8730;3#8730;3#8730;3... is (frac{1 sqrt{13}}{2}).

Generalization of Infinite Radicals

This approach can be generalized to any base (x). Consider the expression √x√x√x√x...

Let:

y sqrt{x cdot sqrt{x cdot sqrt{x ...}}} end{equation>

And by the same logic, we have:

y sqrt{x cdot y} implies y^2 x cdot y implies y^2 - x cdot y 0 implies y(y - x) 0end{equation>

Thus, we obtain two solutions for (y), but we only take the positive one:

y frac{1 sqrt{1 4x}}{2} end{equation>

For the specific case where (x 3), we have:

y frac{1 sqrt{1 12}}{2} frac{1 sqrt{13}}{2} end{equation>

This confirms that the value of the expression #8730;3#8730;3#8730;3#8730;3... is indeed (frac{1 sqrt{13}}{2}).

Conclusion

Infinite nested radicals, such as #8730;3#8730;3#8730;3#8730;3... are beautiful examples of the power and elegance of mathematics. By leveraging basic algebra and a bit of ingenuity, we can solve such problems and gain a deeper understanding of the underlying mathematical principles.

Feel free to ask any questions in the comments below, and I will be happy to help!

For More Information:

Learn more about infinite radicals at MathIsFun. Explore the concept of nested radicals on Wikipedia. Check out Desmos graphing calculator for visualizing these expressions.