$ \frac { 3 ^ { x + 3 } \cdot 12 ^ { x - 3 } } { 2 ^ { 2 x - 6 } , 9 ^ { x } } $

Ask by Brooks West.

Mar 22,2025 09:05

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Answer

The expression simplifies to 1.

Solution

Calculate or simplify the expression $ (3^(x+3) * 12^(x-3)) / (2^(2x-6) * 9^x) $. Simplify the expression by following steps: - step0: Solution: $\frac{\left(3^{x+3}\times 12^{x-3}\right)}{\left(2^{2x-6}\times 9^{x}\right)}$ - step1: Remove the parentheses: $\frac{3^{x+3}\times 12^{x-3}}{2^{2x-6}\times 9^{x}}$ - step2: Factor the expression: $\frac{3^{x+3}\times 12^{x-3}}{2^{2x-6}\times 3^{2x}}$ - step3: Reduce the fraction: $\frac{12^{x-3}}{2^{2x-6}\times 3^{x-3}}$ - step4: Factor the expression: $\frac{2^{2x-6}\times 3^{x-3}}{2^{2x-6}\times 3^{x-3}}$ - step5: Reduce the fraction: $1$ The expression $ \frac {3^{x+3} \cdot 12^{x-3}}{2^{2x-6} \cdot 9^x} $ simplifies to 1.

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Mind Expander

To simplify the expression $ \frac { 3 ^ { x + 3 } \cdot 12 ^ { x - 3 } } { 2 ^ { 2 x - 6 } \cdot 9 ^ { x } } $, we can start by breaking down each part: 1. Notice that $ 12 = 3 \cdot 4 = 3 \cdot 2^2 $, so we can rewrite $ 12^{x-3} = (3 \cdot 2^2)^{x - 3} = 3^{x - 3} \cdot 2^{2(x - 3)} $. 2. Now substitute everything back into the expression: \[ = \frac{3^{x + 3} \cdot (3^{x - 3} \cdot 2^{2(x - 3)})}{2^{2x - 6} \cdot 9^{x}} \] We know that $ 9 = 3^2 $, so $ 9^x = (3^2)^x = 3^{2x} $. 3. Now substituting that back gives: \[ = \frac{3^{x + 3} \cdot 3^{x - 3} \cdot 2^{2(x - 3)}}{2^{2x - 6} \cdot 3^{2x}} \] 4. Combine the terms with $ 3 $ in the numerator: \[ = \frac{3^{(x+3) + (x-3)}}{2^{2(x-3)}} \div \frac{1}{2^{2x-6} \cdot 3^{2x}} \] \[ = \frac{3^{2x}}{2^{2x-6}} \div 2^{2x - 6} \cdot 3^{2x} \] \[ = \frac{3^{2x}}{2^{2x - 6}} \cdot \frac{1}{3^{2x}} = \frac{1}{2^{2x - 6}} \] So, the simplified expression is $ \frac{1}{2^{2x - 6}} $.