Simplify-8-24-8-24-

Question Number 149778 by mathdanisur last updated on 07/Aug/21

Simplify :

\sqrt{8 \sqrt{24 \sqrt{8 \sqrt{24 \dots}}}} = ?

Commented by amin96 last updated on 07/Aug/21

\sqrt{8 \sqrt{24 x}} = x \Rightarrow x^{2} = 8 \sqrt{24 x} \Rightarrow x^{4} = 64 \cdot 24 x

x^{3} = \sqrt{64 \cdot 8 \cdot 3} \Rightarrow x = 4 \cdot 2 \sqrt[3]{3} = 8 \sqrt[3]{3}

Commented by mathdanisur last updated on 07/Aug/21

T h a n k s S e r

Answered by nimnim last updated on 07/Aug/21

S = 8^{\frac{1}{2}} . 24^{\frac{1}{4}} . 8^{\frac{1}{8}} . 24^{\frac{1}{16}} \dots .

= (8^{\frac{1}{2}} . 8^{\frac{1}{8}} \dots \dots \dots) (24^{\frac{1}{4}} . 24^{\frac{1}{16}} \dots \dots)

= 8^{(\frac{1}{2} + \frac{1}{8} + \dots \dots .)} . 24^{(\frac{1}{4} + \frac{1}{16} \dots \dots)} = 8^{(\frac{\frac{1}{2}}{1 - \frac{1}{4}})} . 24^{(\frac{\frac{1}{4}}{1 - \frac{1}{4}})}

= 8^{\frac{2}{3}} . 24^{\frac{1}{3}} = 4.2 . 3^{\frac{1}{3}} = 8 \sqrt[3]{3}

Commented by lyubita last updated on 07/Aug/21

g r e a t \dots

Commented by mathdanisur last updated on 07/Aug/21

T h a n k s S e r