Question

Найдите а³, если
a=
$\sqrt{4 \sqrt{3 \sqrt{4 \sqrt{3 \sqrt{4 \sqrt{...} } } } } }$
​

Answer 1

Воспользуемся такими формулами:

$\sqrt[n]{ {a}^{m} } = {a}^{ \frac{m}{n} }$

$\frac{a}{b} \div c = \frac{a}{b} \times \frac{1}{c}$

${ ({a}^{m}) }^{n} = {a}^{m \times n}$

__

$\sqrt[4]{(...)} = {(...)}^{ \frac{1}{4} }$

$\sqrt[3]{ {(...)}^{ \frac{1}{4} } } = {(...)}^{ \frac{1}{4} \div 3} = {(...)}^{ \frac{1}{12} }$

$\sqrt[4]{ {(...)}^{ \frac{1}{12} } } = {(...)}^{ \frac{1}{12} \div 4} = {(...)}^{ \frac{1}{48} }$

$\sqrt[3]{ {(...)}^{ \frac{1}{48} } } = {(...)}^{ \frac{1}{48} \div 3 } = {(...)}^{ \frac{1}{144} }$

$\sqrt[4]{ {(...)}^{ \frac{1}{144} } } = {(...)}^{ \frac{1}{144} \div 4 } = {(...)}^{ \frac{1}{576} }$

$\sqrt{ {(...)}^{ \frac{1}{576} } } = {(...)}^{ \frac{1}{576} \div 2} = {(...)}^{ \frac{1}{1152} }$

Получается,

$a = {(...)}^{ \frac{1}{1152} }$

${a}^{3} = { ({(...)}^{ \frac{1}{1152} }) }^{3} = {(...)}^{ \frac{1}{1152} \times 3} = {(...)}^{ \frac{1}{384} } = \sqrt[384]{(...)}$