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Chapter 8: Problem 25

Simplify each expression. \(\frac{p^{3}}{2 q} \div \frac{-p}{4 q}\)

Short Answer

Expert verified
The simplified expression is \(-2p^2\).

Step by step solution

01

Rewrite Division as Multiplication

Replace the division part of the expression with multiplication by the reciprocal. The expression \( \frac{p^3}{2q} \div \frac{-p}{4q} \) becomes \( \frac{p^3}{2q} \times \frac{4q}{-p} \).
02

Multiply the Numerators and Denominators

To multiply the fractions, multiply the numerators by each other and the denominators by each other: \( \frac{p^3 \times 4q}{2q \times -p} = \frac{4p^3q}{-2pq} \).
03

Simplify the Fraction

Factor and cancel out common terms in the numerator and denominator. The term \(q\) cancels out, resulting in \( \frac{4p^3}{-2p} = -2p^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Simplification
Fraction simplification is like reducing a recipe to its simplest form. Imagine you have a fraction, and you want to make it as straightforward as possible. This process involves breaking down both the numerator and the denominator until you can't simplify any further.

To simplify a fraction like \( \frac{4p^3}{-2p} \), look for common factors in both parts. In our case, both terms have a factor of \(p\). Cancel out one \(p\) from the numerator and the denominator. This leaves you with \(-2p^2\) since \(4/2 = 2\), and the negative sign remains.

  • Always find the greatest common factor to simplify fractions.
  • Simplification helps in easier calculation and interpretation.
Division of Fractions
Dividing fractions involves a simple trick: multiply by the reciprocal! When you have a fraction division problem, such as \( \frac{p^3}{2q} \div \frac{-p}{4q} \), swap the division for multiplication and flip the fraction you're dividing by.

For example, replace division with multiplication by the reciprocal of \(\frac{-p}{4q}\), turning our expression into \(\frac{p^3}{2q} \times \frac{4q}{-p}\). This swaps the positions of the numerator and the denominator of the second fraction, changing the operation entirely.

  • Remember, "dividing by a fraction is the same as multiplying by its reciprocal".
  • This method often simplifies complicated division problems.
  • Always check the signs - negatives need careful attention!
Multiplication of Fractions
Multiplying fractions is straightforward. It's like multiplying whole numbers but just done twice鈥攐nce for the numerators and once for the denominators. Let's take \(\frac{p^3}{2q} \times \frac{4q}{-p}\) for instance.

First, multiply the numerators: \(p^3 \times 4q\) to get \(4p^3q\). Then, multiply the denominators: \(2q \times -p\) to arrive at \(-2pq\).

Now you have a new fraction: \(\frac{4p^3q}{-2pq}\). Just ensure all multiplication steps are correct, and watch for common factors to simplify later!

  • Multiply tops with tops and bottoms with bottoms!
  • Be mindful of negative signs.
  • This method forms the basis for more complex algebraic operations.

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Most popular questions from this chapter

Graph \(y \leq \sqrt{x+1}\)

Solve each equation or inequality. Check your solutions. $$ 7-\frac{2}{b}<\frac{5}{b} $$

Identify each table of values as a type of function. A. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-5 & {7} \\ \hline-3 & {5} \\ \hline-1 & {3} \\ \hline 0 & {2} \\ \hline 1 & {3} \\ \hline 3 & {5} \\ \hline 5 & {7} \\ \hline 7 & {9} \\ \hline\end{array}\) B. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-5 & {24} \\ \hline-3 & {8} \\ \hline-1 & {0} \\ \hline 0 & {-1} \\ \hline 1 & {0} \\ \hline 3 & {8} \\ \hline 5 & {24} \\ \hline 7 & {48} \\ \hline\end{array}\) C. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-1.3 & {-1} \\ \hline-1.7 & {-1} \\ \hline 0 & {1} \\ \hline 0.8 & {1} \\ \hline 0.9 & {1} \\ \hline 0.9 & {1} \\ \hline 1 & {2} \\ \hline 1.5 & {2} \\ \hline 2.3 & {3} \\\ \hline\end{array}\) D. \(\begin{array}{|c|c|}\hline x & {f(x)} \\ \hline-5 & {\text { undefined }} \\\ \hline-3 & {\text { undefined }} \\ \hline-1 & {\text { undefined }} \\\ \hline 0 & {0} \\ \hline 1 & {1} \\ \hline 4 & {2} \\ \hline 9 & {3} \\\ \hline 16 & {4} \\ \hline\end{array}\)

Factor each polynomial. $$ x^{2}+3 x+2 $$

Solve each equation or inequality. Check your solutions. $$ 5+\frac{1}{t}>\frac{16}{t} $$
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