Given that t = \(2 ^{-x}\), find \(2 ^{x + 1}\) in terms of t.

Given that t = \(2 ^{-x}\), find \(2 ^{x + 1}\) in terms of t. 

Answer Details

We can start by using the laws of exponents to rewrite \(2^{x+1}\) in terms of \(2^{-x}\), which is given by \(t\). Recall that \(a^{m+n} = a^m \times a^n\) and \(a^{-n} = \frac{1}{a^n}\). Therefore, we have: $$2^{x+1} = 2^x \times 2^1 = 2^x \times 2 = 2 \times 2^x$$ Next, we can substitute \(2^{-x} = t\) into this expression: $$2 \times 2^x = 2 \times \frac{1}{t} = \frac{2}{t}$$ So the answer is: \(\frac{2}{t}\).