Given that \(\sqrt {7\sqrt {7\sqrt {7\sqrt {7\sqrt {7 \ldots \ldots \ldots \ldots \ldots } } } } } = {117649^a}\) Then find the value of a.

A 1/2

B 1/3

C 1/4

D 1/5

E None of these

Correct Answer: Option E

Laws of indices:

(a^m)ⁿ = a^{m × n}

a^m × aⁿ = a^m+n

a^m ÷ aⁿ = a^m-n

LHS = \(\sqrt {7\sqrt {7\sqrt {7\sqrt {7\sqrt {7 \ldots \ldots \ldots \ldots \ldots } } } } }\) and RHS = 117649a

Let LHS = y, then

\(y = \sqrt {7\sqrt {7\sqrt {7\sqrt {7\sqrt {7 \ldots \ldots \ldots \ldots \ldots } } } } } \;\)

Or y = √(7y) Squaring both sides, we get

⇒ y² = 7y

⇒ y² – 7y = 0

⇒ y (y – 7) = 0

Since y is a non-zero number, thus y = 7

Now, RHS = 7

Or, 117649^a = 7

⇒ (7⁶)^a = 7

⇒ 7^6a = 7

Comparing both sides, we obtain

6a = 1

Or, a = 1/6