Verified Answer :

  • Distribute the denominator 8 across the terms in the numerator: \(\frac{32y^9}{8} - \frac{24y^3}{8}\).
  • Divide the coefficients 32 and 24 by 8 to get 4 and 3, respectively.
  • Combine the terms to reach the final simplified expression of \(\boxed{4y^9 - 3y^3}\).

Explanation

  1. Problem Analysis

We are given the rational expression \(\frac{32y^9 - 24y^3}{8}\). Our goal is to simplify this expression by dividing both terms in the numerator by the denominator, which is 8. Since 8 is a common divisor for both 32 and 24, we can distribute the division across the subtraction in the numerator.

  1. Simplification Process

To simplify the expression, we distribute the denominator 8 to each term in the numerator:
\[\frac{32y^9 - 24y^3}{8} = \frac{32y^9}{8} - \frac{24y^3}{8}\]
Next, we divide the coefficients:
For the first term: \(\frac{32}{8} = 4\), so \(\frac{32y^9}{8} = 4y^9\).
For the second term: \(\frac{24}{8} = 3\), so \(\frac{24y^3}{8} = 3y^3\).
Combining these results, we get the simplified expression:
\[4y^9 - 3y^3\]

  1. Conclusion

The expression simplifies to \(4y^9 - 3y^3\). This is the final form of the expression after dividing each term of the numerator by the denominator. Thus, the simplified result is \(\boxed{4y^9 - 3y^3}\).

Examples

In manufacturing, you might have a total cost function for producing \(y\) units, such as \(\frac{32y^9 - 24y^3}{8}\) dollars. Simplifying this expression helps determine the average or scaled cost per batch, which in this case becomes \(4y^9 - 3y^3\) dollars. Understanding how to divide expressions allows for easier cost estimation and budget management as production volume changes.

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