4.2. \( \frac{2^{2 x-1} \cdot 4^{x+1} \cdot 2}{16^{x}} \)

Simplify the expression by following steps: - step0: Solution: \(2\left(2x-1\right)\) - step1: Rearrange the terms: \(4x-2\) Calculate or simplify the expression \( 4**(x+1) \). Simplify the expression by following steps: - step0: Solution: \(4\left(x+1\right)\) - step1: Rearrange the terms: \(4x+4\) Calculate or simplify the expression \( 16**(x) \). Simplify the expression by following steps: - step0: Solution: \(16x\) Calculate or simplify the expression \( 2**(4*x-1) / 2**(16*x) \). Simplify the expression by following steps: - step0: Solution: \(2\times \frac{\left(4x-1\right)}{2}\times 16x\) - step1: Remove the parentheses: \(2\times \frac{4x-1}{2}\times 16x\) - step2: Multiply the terms: \(32\times \frac{4x-1}{2}\times x\) - step3: Reduce the fraction: \(16\left(4x-1\right)x\) - step4: Multiply the terms: \(\left(64x-16\right)x\) - step5: Apply the distributive property: \(64x\times x-16x\) - step6: Multiply the terms: \(64x^{2}-16x\) Let's simplify the expression step by step: The original expression is: \[ \frac{2^{2x-1} \cdot 4^{x+1} \cdot 2}{16^{x}} \] ### Step 1: Rewrite the bases We can express all terms with base 2: - \(4 = 2^2\), so \(4^{x+1} = (2^2)^{x+1} = 2^{2(x+1)} = 2^{2x + 2}\) - \(16 = 2^4\), so \(16^{x} = (2^4)^{x} = 2^{4x}\) Now, substituting these into the expression gives: \[ \frac{2^{2x-1} \cdot 2^{2x + 2} \cdot 2}{2^{4x}} \] ### Step 2: Combine the exponents in the numerator Using the property of exponents \(a^m \cdot a^n = a^{m+n}\): \[ 2^{(2x-1) + (2x + 2) + 1} = 2^{2x - 1 + 2x + 2 + 1} = 2^{4x + 2} \] ### Step 3: Simplify the entire expression Now we have: \[ \frac{2^{4x + 2}}{2^{4x}} = 2^{(4x + 2) - 4x} = 2^{2} \] ### Final Result Thus, the simplified expression is: \[ 2^{2} = 4 \] So, the final answer is: \[ \boxed{4} \]