Let's simplify the expression T = \(\frac{4R_2}{R_1^{-1} + R_2^{-1} + 4R_3^{-1}}\) step by step:

1. We can rewrite the denominator as \(R_1^{-1} + R_2^{-1} + 4R_3^{-1} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{4}{R_3}\)
2. We can rewrite the expression as \(T = \frac{4R_2}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{4}{R_3}}\)
3. We can multiply both the numerator and denominator by \(R_1 R_2 R_3\) to get rid of the fractions in the denominator: \(T = \frac{4R_2(R_1 R_2 R_3)}{(R_2 R_3 + R_1 R_3 + 4R_1 R_2)}\)
4. Simplifying further, we get \(T = \frac{4R_1 R_2 R_3}{(R_2 R_3 + R_1 R_3 + 4R_1 R_2)}\)