Questions: Find the quotient and simplify the result. (x-3)(x+4)/4x^9 ÷ (5x-15)/20x^10 (x-3)(x+4)/4x^9 ÷ (5x-15)/20x^10 =

Find the quotient and simplify the result. (x-3)(x+4)/4x^9 ÷ (5x-15)/20x^10 (x-3)(x+4)/4x^9 ÷ (5x-15)/20x^10 =
Transcript text: Find the quotient and simplify the result. \[ \begin{array}{l} \frac{(x-3)(x+4)}{4 x^{9}} \div \frac{5 x-15}{20 x^{10}} \\ \frac{(x-3)(x+4)}{4 x^{9}} \div \frac{5 x-15}{20 x^{10}}= \end{array} \]
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Solution

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Solution Steps

To solve the given problem, we need to divide two rational expressions. The division of fractions is equivalent to multiplying by the reciprocal. Therefore, we will multiply the first fraction by the reciprocal of the second fraction. After that, we will simplify the expression by canceling out common factors in the numerator and the denominator.

Step 1: Define the Expressions

We start with the two rational expressions: \[ \frac{(x-3)(x+4)}{4x^9} \quad \text{and} \quad \frac{5x-15}{20x^{10}}. \]

Step 2: Rewrite the Division as Multiplication

To divide the first expression by the second, we multiply by the reciprocal of the second expression: \[ \frac{(x-3)(x+4)}{4x^9} \div \frac{5x-15}{20x^{10}} = \frac{(x-3)(x+4)}{4x^9} \cdot \frac{20x^{10}}{5x-15}. \]

Step 3: Simplify the Expression

Now, we can simplify the expression: \[ \frac{(x-3)(x+4) \cdot 20x^{10}}{4x^9 \cdot (5x-15)}. \] Notice that \(5x - 15\) can be factored as \(5(x - 3)\): \[ = \frac{(x-3)(x+4) \cdot 20x^{10}}{4x^9 \cdot 5(x-3)}. \] We can cancel the common factor \((x-3)\): \[ = \frac{20(x+4)x^{10}}{20x^9} = \frac{(x+4)x^{10}}{x^9}. \] This simplifies to: \[ = x(x + 4). \]

Final Answer

Thus, the simplified result is: \[ \boxed{x(x + 4)}. \]

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