What is sqrt(7sqrt(7sqrt7)) ?

Question

What is

\sqrt{7 \sqrt{7 \sqrt{7}}}

?

Answer 1

Lets asume:
$y = \sqrt{7 \sqrt{7 \sqrt{7}}}$ ,
$y^{2} = 7 \sqrt{7 \sqrt{7}}$ ,
$y^{4} = 7^{2} \cdot 7 \sqrt{7} = 7^{3} \sqrt{7}$ ,
$y^{8} = 7^{6} \cdot 7 = 7^{7}$
So $y = 7^{\frac{7}{8}} = 5.4886$ . In general, if N is the number of 7's under the square root then $y = 7^{\frac{(2^{N}) - 1}{2^{N}}}$ . In this case N=3 so $y = 7^{\frac{2^{3} - 1}{2^{3}}} = 7^{\frac{7}{8}}$ . For N=4, $y = 7^{\frac{15}{16}} = 6.1984$ , as N approaches infinity y approaches 7. So for 7's repeated under square roots indefinitely the answer is 7.
Another way of solving it is to put $y = \sqrt{7 (\sqrt{7} (\sqrt{7} \dots}$ So $y^{2} = 7 \sqrt{7} (\sqrt{7} \sqrt{7} \dots = 7 y$ , so since $y \neq 0, y = 7$ (dividing through by y).

What is sqrt(7sqrt(7sqrt7)) ?

Want to know more about Irrational numbers?

Expert Answer

Not exactly what you’re looking for?

You might be interested in

New questions