Question

Simplify-

$\sqrt{6 \sqrt{6 \sqrt{6 \sqrt{6...} } } }$
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Answer 1

HEY MATE HERE IS YOUR ANSWER--

Let us assume : x= root(6+root(6 + root(6 + ………

So,

x= root(6+x)

x^2=6+x

x^2-x-6=0

(x-3)(x+2)=0

So, x=3 or -2.

But taking -2 as the answer would mean we are taking root of (-2) . Considering the fact that x belongs to the set of real numbers , we discard the solution x=-2.

So the answer is x =3.

Answer 2

▶ Question :-

→ Simplify :-

$\sf \sqrt{6 \sqrt{6 \sqrt{6 \sqrt{6 ... \infty } } } } .$

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▶ Answer :-

→ 6.

▶ Step-by-step explanation :-



$\sf Let \: x = \sqrt{6 \sqrt{6 \sqrt{6 \sqrt{6.... \infty } } } } . \\ \\ \sf \implies x = \sqrt{6 \bigg( \sqrt{6 \sqrt{6 \sqrt{6... \infty } } } \bigg) } \\ \\ \sf \implies x = \sqrt{6 x} . \\ \\ \{ \tt squaring \: both \: side \} \\ \\ \sf \implies {x}^{2} = 6 x.$


⇒ x² - 6x = 0 .

⇒ x( x - 6 ) = 0 .

⇒ x - 6 = 0/x .

⇒ x - 6 = 0 .

∴ x = 6 .



$\orange{ \boxed{ \sf \therefore \sqrt{6 \sqrt{6 \sqrt{6 \sqrt{6... \infty } } } } = 6.}}$

Hence, it is solved.