To divide the fractions, we start with the expression:
\[ \frac{9}{4x^2} \div \frac{6x}{16x^4} \]
When dividing by a fraction, we multiply by its reciprocal:
\[ \frac{9}{4x^2} \times \frac{16x^4}{6x} \]
Now we can simplify the expression before we multiply:
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Multiply the numerators and denominators: \[ = \frac{9 \times 16x^4}{4x^2 \times 6x} \]
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Simplifying the numerator: \[ 9 \times 16 = 144 \] So the numerator becomes \(144x^4\).
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Simplifying the denominator: \[ 4 \times 6 = 24 \quad \text{and} \quad x^2 \times x = 6x^3 \] So the denominator becomes \(24x^3\).
Thus, we have:
\[ \frac{144x^4}{24x^3} \]
- Dividing both the numerator and denominator by 24: \[ = \frac{144 \div 24}{24 \div 24} \cdot \frac{x^4}{x^3} = \frac{6}{1} \cdot x^{4-3} = 6x \]
Thus, the simplified answer is:
E. 6x
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