Question

(d) \( \frac{10^{x} \cdot 25^{x-1} \cdot 2}{50^{x+1}} \)

Ask by Henry Sherman. in South Africa
Feb 25,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The simplified form of the expression is \( 5^{x-4} \).

Solution

Calculate or simplify the expression \( 10^{x} \cdot 25^{x-1} \cdot 2/50^{x+1} \). Simplify the expression by following steps: - step0: Solution: \(\frac{10^{x}\times 25^{x-1}\times 2}{50^{x+1}}\) - step1: Factor the expression: \(\frac{10^{x}\times 25^{x-1}\times 2}{2\times 10^{x}\times 5^{2+x}}\) - step2: Reduce the fraction: \(\frac{25^{x-1}}{5^{2+x}}\) - step3: Factor the expression: \(\frac{5^{2x-2}}{5^{2+x}}\) - step4: Divide the numbers: \(5^{2x-2-\left(2+x\right)}\) - step5: Subtract the terms: \(5^{x-4}\) The simplified form of the expression \( \frac{10^{x} \cdot 25^{x-1} \cdot 2}{50^{x+1}} \) is \( 5^{x-4} \).

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To simplify the expression \( \frac{10^{x} \cdot 25^{x-1} \cdot 2}{50^{x+1}} \), we can first rewrite the bases in terms of prime factors. We know that \( 10 = 2 \cdot 5 \), \( 25 = 5^2 \), and \( 50 = 2 \cdot 5^2 \). We can rewrite the expression: - \( 10^x = (2 \cdot 5)^x = 2^x \cdot 5^x \) - \( 25^{x-1} = (5^2)^{x-1} = 5^{2(x-1)} = 5^{2x - 2} \) - \( 50^{x+1} = (2 \cdot 5^2)^{x+1} = 2^{x+1} \cdot 5^{2(x+1)} = 2^{x+1} \cdot 5^{2x + 2} \) Putting this all together, we have: \[ \frac{2^x \cdot 5^x \cdot 5^{2x - 2} \cdot 2}{2^{x+1} \cdot 5^{2x + 2}} \] Now combine the powers of 5 in the numerator: \[ = \frac{2^x \cdot 5^{x + 2x - 2} \cdot 2}{2^{x+1} \cdot 5^{2x + 2}} = \frac{2^{x + 1} \cdot 5^{3x - 2}}{2^{x + 1} \cdot 5^{2x + 2}} \] Now we can cancel \( 2^{x + 1} \) from the numerator and denominator: \[ = \frac{5^{3x - 2}}{5^{2x + 2}} = 5^{(3x - 2) - (2x + 2)} = 5^{x - 4} \] So the final simplified result is: \[ 5^{x - 4} \]

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