To divide the fractions \(\frac{25}{4x}\) by \(\frac{10x^2}{8x^4}\), we can multiply by the reciprocal of the second fraction.
Here’s how you can do it step by step:
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Rewrite the division as multiplication by the reciprocal: \[ \frac{25}{4x} \div \frac{10x^2}{8x^4} = \frac{25}{4x} \times \frac{8x^4}{10x^2} \]
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Multiply the numerators and the denominators: \[ = \frac{25 \times 8x^4}{4x \times 10x^2} \]
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Simplifying the numerators and denominators: \[ = \frac{200x^4}{40x^3} \]
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Now simplify the fraction \( \frac{200}{40} \) and \( \frac{x^4}{x^3} \): \[ = \frac{200 \div 40}{40 \div 40} \frac{x^4}{x^3} = \frac{5}{1} x^{4-3} = 5x \]
Thus, the simplified result of dividing the fractions is: \[ \boxed{5x} \]
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