Question

b.) \( \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} \)

Ask by Weaver Valdez. in South Africa
Jan 23,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The simplified form of the expression is 2.

Solution

To simplify the expression: \[ \frac{2^{x+2} - 2^{x+3}}{2^{x+1} - 2^{x+2}} \] **Step 1: Factor Out Common Terms** Start by factoring out the smallest power of 2 in both the numerator and the denominator. - **Numerator:** \[ 2^{x+2} - 2^{x+3} = 2^{x+2}(1 - 2) = 2^{x+2}(-1) = -2^{x+2} \] - **Denominator:** \[ 2^{x+1} - 2^{x+2} = 2^{x+1}(1 - 2) = 2^{x+1}(-1) = -2^{x+1} \] **Step 2: Simplify the Expression** Now, substitute the factored forms back into the original expression: \[ \frac{-2^{x+2}}{-2^{x+1}} = \frac{2^{x+2}}{2^{x+1}} \] **Step 3: Use the Laws of Exponents** When dividing like bases, subtract the exponents: \[ \frac{2^{x+2}}{2^{x+1}} = 2^{(x+2)-(x+1)} = 2^1 = 2 \] **Final Answer:** \[ 2 \]

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The Deep Dive

To simplify the expression \( \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} \), let's start with the numerator: \[ 2^{x+2} - 2^{x+3} = 2^{x+2}(1 - 2) = 2^{x+2}(-1) = -2^{x+2} \] Next, let's simplify the denominator: \[ 2^{x+1} - 2^{x+2} = 2^{x+1}(1 - 2) = 2^{x+1}(-1) = -2^{x+1} \] Now we can rewrite the entire expression: \[ \frac{-2^{x+2}}{-2^{x+1}} = \frac{2^{x+2}}{2^{x+1}} \] When we simplify this, we use the property of exponents: \[ \frac{2^{x+2}}{2^{x+1}} = 2^{(x+2)-(x+1)} = 2^{1} = 2 \] Thus, the simplified expression is: \[ \boxed{2} \]

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