Question
implify the following \( \frac{4^{x} \cdot 2^{x-2}}{8^{x-1}} \)
Ask by Gough Owen. in South Africa
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is 2.
Solution
Step 1: Express 4 and 8 as powers of 2.
• 4 = 2², so 4ˣ = (2²)ˣ = 2^(2x).
• 8 = 2³, so 8^(x-1) = (2³)^(x-1) = 2^(3x-3).
Step 2: Rewrite the original expression in terms of base 2.
(2^(2x) · 2^(x-2)) / 2^(3x-3)
Step 3: Simplify the numerator using the property a^m · a^n = a^(m+n).
2^(2x) · 2^(x-2) = 2^(2x + x - 2) = 2^(3x - 2)
Step 4: Now simplify the fraction using the property a^m / a^n = a^(m-n).
2^(3x - 2) / 2^(3x - 3) = 2^[(3x - 2) - (3x - 3)] = 2^(1) = 2
So, the simplified form of the expression is 2.
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Simplify this solution Beyond the Answer
First, let's re-write the expression in terms of powers of 2. We know that \( 4 = 2^2 \) and \( 8 = 2^3 \). So we can express everything as follows: \[ 4^x = (2^2)^x = 2^{2x} \] \[ 8^{x-1} = (2^3)^{x-1} = 2^{3(x-1)} = 2^{3x - 3} \] Now, substitute these into the original expression: \[ \frac{4^{x} \cdot 2^{x-2}}{8^{x-1}} = \frac{2^{2x} \cdot 2^{x-2}}{2^{3x-3}} \] Combine the powers in the numerator: \[ 2^{2x} \cdot 2^{x-2} = 2^{2x + x - 2} = 2^{3x - 2} \] So now the expression becomes: \[ \frac{2^{3x - 2}}{2^{3x - 3}} = 2^{(3x - 2) - (3x - 3)} = 2^{1} = 2 \] Thus, the simplified form of the original expression is: \[ \boxed{2} \]
