Simplify the complex fraction.
$\frac{\frac{16 c^4 d^3}{7 c}}{\frac{4 c d^4}{d^2}}$
Answer (1)
Rewrite the complex fraction as a division problem.
Change the division to multiplication by inverting the second fraction.
Multiply the numerators and denominators.
Simplify the expression by canceling common factors, resulting in 7 4 c 2 d .
Explanation
Understanding the Problem We are asked to simplify the complex fraction d 2 4 c d 4 7 c 16 c 4 d 3 . This involves simplifying the numerator and denominator separately, and then dividing the simplified numerator by the simplified denominator.
Rewriting as Division First, we rewrite the complex fraction as a division problem: d 2 4 c d 4 7 c 16 c 4 d 3 = 7 c 16 c 4 d 3 ÷ d 2 4 c d 4
Changing to Multiplication Next, we change the division to multiplication by inverting the second fraction: 7 c 16 c 4 d 3 ÷ d 2 4 c d 4 = 7 c 16 c 4 d 3 ⋅ 4 c d 4 d 2
Multiplying Numerators and Denominators Now, we multiply the numerators and denominators: 7 c 16 c 4 d 3 ⋅ 4 c d 4 d 2 = 7 c ( 4 c d 4 ) 16 c 4 d 3 d 2 = 28 c 2 d 4 16 c 4 d 5
Simplifying the Expression We simplify the expression by dividing the numerator and denominator by their greatest common factor. First, we simplify the coefficients: 28 16 = 7 4 Then, we simplify the variables: c 2 c 4 = c 4 − 2 = c 2 d 4 d 5 = d 5 − 4 = d
Final Simplified Expression Finally, we write the simplified expression: 28 c 2 d 4 16 c 4 d 5 = 7 4 c 2 d
Examples
Complex fractions might seem abstract, but they appear in various real-world scenarios. For instance, when calculating the average speed of a car that travels different distances at different speeds, you often encounter complex fractions. Similarly, in electrical engineering, when dealing with parallel circuits, you might need to simplify complex fractions to find the equivalent resistance. Understanding how to simplify these fractions helps in making these calculations more manageable and understandable.
