Questions: Divide and simplify. 49 x^4 / 7 x^5 ÷ 7 x^5 / 3 x^6 The simplified quotient is.

Solution Steps

To solve the given problem, we need to divide two fractions and simplify the result. The division of fractions can be handled by multiplying the first fraction by the reciprocal of the second fraction. After that, we simplify the expression by canceling out common terms in the numerator and the denominator.

We start with the expression to simplify: \[ \frac{49 x^{4}}{7 x^{5}} \div \frac{7 x^{5}}{3 x^{6}} \]

We can rewrite the division of fractions as multiplication by the reciprocal: \[ \frac{49 x^{4}}{7 x^{5}} \times \frac{3 x^{6}}{7 x^{5}} \]

Now, we simplify each fraction:

1. For the first fraction: \[ \frac{49 x^{4}}{7 x^{5}} = \frac{49}{7} \cdot \frac{x^{4}}{x^{5}} = 7 \cdot \frac{1}{x} = \frac{7}{x} \]
2. For the second fraction: \[ \frac{3 x^{6}}{7 x^{5}} = \frac{3}{7} \cdot \frac{x^{6}}{x^{5}} = \frac{3}{7} \cdot x = \frac{3x}{7} \]

Now we multiply the simplified fractions: \[ \frac{7}{x} \times \frac{3x}{7} = \frac{7 \cdot 3x}{x \cdot 7} = \frac{21x}{7x} \]

We can simplify \(\frac{21x}{7x}\): \[ \frac{21}{7} = 3 \]

Final Answer