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# Dots

$$\sqrt{\sqrt{\sqrt{\sqrt{4.\underbrace {00\dots}_{300}3}}}}$$ = ?

Note by Yannawat Praserttham
2 years, 1 month ago

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What are you looking for exactly? The simplest form of the answer? The first few digits of the number?

If you're looking for the first few digits of the number, try $$(1+x)^n \approx 1 + nx$$. for small $$x$$.

- 2 years, 1 month ago

The simplest form of the answer.

- 2 years, 1 month ago

Hint: convert decimal number into an improper fraction.

- 2 years, 1 month ago

I did that, but i don't know what to do next.

- 2 years, 1 month ago

Show me what you've done so far.

- 2 years, 1 month ago

$$4.\underbrace {00\dots}_{300}3$$ = $$\frac{(4 \times 10^{301})+3}{10^{301}}$$

= $$\frac{4 \times 10^{301}}{10^{301}}+\frac{3}{10^{301}}$$

= $$4+\frac{3}{10^{301}}$$

- 2 years, 1 month ago

Combine it into a single fraction, you have $$\dfrac{4\cdot 10^{301} + 3}{10^{301}}$$. Notice that neither e numerator nor the denominator is not a perfect square (do you know how to prove that?), so your final answer is simply $$\left( \dfrac{4\cdot 10^{301} + 3}{10^{301}} \right)^{1/16}$$.

- 2 years, 1 month ago