Dots

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

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 $x$ .

Pi Han Goh - 5 years, 4 months ago

The simplest form of the answer.

Yannawat Praserttham - 5 years, 4 months ago

Hint: convert decimal number into an improper fraction.

Pi Han Goh - 5 years, 4 months ago

@Pi Han Goh – I did that, but i don't know what to do next.

Yannawat Praserttham - 5 years, 4 months ago

@Yannawat Praserttham – Show me what you've done so far.

Pi Han Goh - 5 years, 4 months ago

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

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

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

Yannawat Praserttham - 5 years, 4 months ago

@Yannawat Praserttham – 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 $\left( \dfrac{4\cdot 10^{301} + 3}{10^{301}} \right)^{1/16}$ .

Pi Han Goh - 5 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$