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maxkor
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How prove that $2^{2^{\sqrt3}}>10$ without a calculator?
maxkor said:How prove that $2^{2^{\sqrt3}}>10$ without a calculator?
Prove It said:$\displaystyle \begin{align*} \sqrt{1} < \sqrt{3} &< \sqrt{4} \\ 1 < \sqrt{3} &< 2 \\ 2^1 < 2^{\sqrt{3}} &< 2^2 \\ 2 < 2^{\sqrt{3}} &< 4 \\ 2^2 < 2^{2^{\sqrt{3}}} &< 2^4 \\ 4 < 2^{2^{\sqrt{3}}} &< 16 \end{align*}$
That may have been a start at least...
maxkor said:and what next?
My first thought is to estimate the difficulty of the problem by using a calculator to find the numerical value of $2^{2^{\sqrt3}}$. The result I get is that (to six decimal places) it is equal to $10.000478$. That is so close to $10$ that a very accurate calculation will be needed to prove this inequality, and I don't see any way of doing it by hand.maxkor said:How prove that $2^{2^{\sqrt3}}>10$ without a calculator?
To solve this without a calculator, you can use logarithms. Specifically, you can take the logarithm of both sides of the inequality to get $2^{\sqrt3}>2$. Then, using the property of logarithms, you can rewrite this as $\sqrt3>1$, which is true. Therefore, the original inequality holds and $2^{2^{\sqrt3}}>10$.
Logarithms allow us to solve exponential equations without using a calculator. In this case, we can use logarithms to simplify the original inequality and prove that it is true.
Yes, there are other methods that can be used to solve this problem without a calculator. For example, you can use the properties of exponents to rewrite $2^{2^{\sqrt3}}$ as $(2^{\sqrt3})^2$, which can be simplified to $4^{\sqrt3}$. Then, using the definition of logarithms, you can rewrite $4^{\sqrt3}$ as $e^{\sqrt3\ln4}$, which can be approximated using the natural logarithm, $e$, and basic arithmetic operations.
No, it is not possible to solve this problem without any mathematical tools or techniques. The given inequality involves exponents, which cannot be simplified without using logarithms or other mathematical concepts.
Yes, this problem can be easily solved using a calculator. You can simply input the expression $2^{2^{\sqrt3}}$ and compare it to 10. The calculator will show that $2^{2^{\sqrt3}}>10$, confirming the given inequality.