Lorentz factor = 10^6, solving for Beta

[demoncore]

Homework Statement

A particle moves such that its relativistic factor γ equals one million. Find β (=v/c).
Give answer in the form β = 0.999..., with correct number of nines before first non-nine digit. Do not use a calculator

Homework Equations

(1-β^2)^(-1/2) = 10^6

The Attempt at a Solution

I've messed around algebraically in a few ways, and tried the binomial approximation, but can't seem to find a way to deal with the β^2 without a calculator.

 

[TSny]

Try to manipulate γ = 10⁶ into an expression for β² of the form β² = 1-ε where ε is a very small number that you will determine.

Then β = [itex]\sqrt{1-ε}[/itex]. Expand this to first order in ε.