2.2 Given: \( \frac{4^{x-1}+4^{x+1}}{17.12^{x}} \) 2.2.1 \( \quad \) Simplify the expression fully. \( 2.2 .2 \quad \) If \( 3^{-x}=4 t \), express \( \frac{4^{x-1}+4^{x+1}}{17.12^{x}} \) in terms of \( t \).

**Step 1. Simplify the expression** We start with the expression \[ \frac{4^{x-1}+4^{x+1}}{17\cdot 12^{x}}. \] Factor the numerator by factoring out \(4^{x-1}\): \[ 4^{x-1} + 4^{x+1} = 4^{x-1}\left(1 + 4^2\right) = 4^{x-1}\left(1 + 16\right)= 17\cdot 4^{x-1}. \] So, the expression becomes: \[ \frac{17\cdot 4^{x-1}}{17\cdot 12^{x}}. \] Cancel the factor of \(17\) from the numerator and denominator: \[ \frac{4^{x-1}}{12^{x}}. \] Next, write \(4^{x-1}\) as \(\frac{4^x}{4}\): \[ \frac{4^x}{4\cdot 12^{x}} = \frac{4^x}{12^x} \cdot \frac{1}{4}. \] Since \(\frac{4^x}{12^x} = \left(\frac{4}{12}\right)^x\), and \(\frac{4}{12}=\frac{1}{3}\), we have: \[ \left(\frac{1}{3}\right)^x \cdot \frac{1}{4}. \] Thus, the simplified form is: \[ \frac{1}{4}\left(\frac{1}{3}\right)^x. \] --- **Step 2. Express the simplified expression in terms of \(t\)** We are given that \[ 3^{-x} = 4t. \] Notice that \[ \left(\frac{1}{3}\right)^x = 3^{-x}. \] Substitute \(3^{-x}\) by \(4t\) in the simplified expression: \[ \frac{1}{4}\left(\frac{1}{3}\right)^x = \frac{1}{4}\cdot 3^{-x} = \frac{1}{4}\cdot (4t) = t. \] Thus, expressed in terms of \(t\), the expression is: \[ t. \]