Questions: (5a^2+27a+28)/(a^4-256) ÷ (25a^2+70a+4a)/(a^2-4a)

(5a^2+27a+28)/(a^4-256) ÷ (25a^2+70a+4a)/(a^2-4a)
Transcript text: $\frac{5 a^{2}+27 a+28}{a^{4}-256} \div \frac{25 a^{2}+70 a+4 a}{a^{2}-4 a}$
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Solution

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Solution Steps

To solve the given expression, we need to follow these steps:

  1. Simplify the numerator and the denominator of each fraction separately.
  2. Perform the division of the two fractions by multiplying the first fraction by the reciprocal of the second fraction.
  3. Simplify the resulting expression if possible.
Step 1: Simplify the Numerator and Denominator of Each Fraction

First, we simplify the numerator and denominator of each fraction separately:

  • For the first fraction: \[ \frac{5a^2 + 27a + 28}{a^4 - 256} \]
  • For the second fraction: \[ \frac{25a^2 + 74a}{a^2 - 4a} \]
Step 2: Perform the Division by Multiplying by the Reciprocal

Next, we perform the division of the two fractions by multiplying the first fraction by the reciprocal of the second fraction: \[ \frac{5a^2 + 27a + 28}{a^4 - 256} \div \frac{25a^2 + 74a}{a^2 - 4a} = \frac{5a^2 + 27a + 28}{a^4 - 256} \times \frac{a^2 - 4a}{25a^2 + 74a} \]

Step 3: Simplify the Resulting Expression

We simplify the resulting expression: \[ \frac{(5a^2 + 27a + 28)(a^2 - 4a)}{(a^4 - 256)(25a^2 + 74a)} \] After simplification, we get: \[ \frac{5a + 7}{25a^3 + 74a^2 + 400a + 1184} \]

Final Answer

\(\boxed{\frac{5a + 7}{25a^3 + 74a^2 + 400a + 1184}}\)

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