Questions: Multiply. [ (20 x+20)/(20 x+60) cdot (5 x+15)/(4 x^2-4) ]

Multiply. [ (20 x+20)/(20 x+60) cdot (5 x+15)/(4 x^2-4) ]
Transcript text: rk Section 6.1 Multiply. \[ \frac{20 x+20}{20 x+60} \cdot \frac{5 x+15}{4 x^{2}-4} \]
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Solution

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Solution Steps

To multiply the given fractions, we first simplify each fraction by factoring the numerators and denominators. Then, we multiply the simplified fractions and simplify the resulting fraction if possible.

Solution Approach
  1. Factor the numerators and denominators of each fraction.
  2. Cancel out any common factors between the numerators and denominators.
  3. Multiply the remaining factors to get the final simplified fraction.
Step 1: Simplifying the First Fraction

We start with the first fraction: \[ \frac{20x + 20}{20x + 60} \] Factoring out the common terms, we get: \[ \frac{20(x + 1)}{20(x + 3)} = \frac{x + 1}{x + 3} \]

Step 2: Simplifying the Second Fraction

Next, we simplify the second fraction: \[ \frac{5x + 15}{4x^2 - 4} \] Factoring gives us: \[ \frac{5(x + 3)}{4(x^2 - 1)} = \frac{5(x + 3)}{4(x - 1)(x + 1)} \]

Step 3: Multiplying the Simplified Fractions

Now we multiply the two simplified fractions: \[ \frac{x + 1}{x + 3} \cdot \frac{5(x + 3)}{4(x - 1)(x + 1)} \] Cancelling the common factors \((x + 1)\) and \((x + 3)\), we have: \[ \frac{5}{4(x - 1)} \]

Final Answer

Thus, the final simplified result is: \[ \boxed{\frac{5}{4(x - 1)}} \]

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