Problem 31 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(-2p\).

Step 1: Rewrite the Division as Multiplication

To simplify the expression \( \frac{\frac{p^{3}}{2q}}{-\frac{p^{2}}{4q}} \), rewrite the division of fractions as multiplication by the reciprocal. This becomes: \[ \frac{p^{3}}{2q} \times \left(-\frac{4q}{p^{2}}\right) \]

Step 2: Multiply the Numerators and Denominators

Perform the multiplication by multiplying the numerators together and the denominators together: \[ \frac{p^{3} \times -4q}{2q \times p^{2}} \] This results in the expression \(-\frac{4p^{3}q}{2qp^{2}}\).

Step 3: Simplify the Expression

Cancel out common factors in the numerator and denominator. Both the numerator and the denominator have a \(q\) and \(p^{2}\). This simplifies to: \[ -\frac{4p \cancel{p^2} \cancel{q}}{2 \cancel{q} \cancel{p^2}} \Rightarrow -\frac{4p}{2} \].

Step 4: Divide Remaining Terms

Divide \(-4p\) by \(2\) to simplify the expression: \[ -2p \].

Key Concepts

Division of Fractions

Understanding division is key in dealing with fractions. When you see a fraction divided by another fraction, it can seem confusing. But there's a neat trick to simplify it. Instead of dividing directly, you change it into a multiplication problem. This involves flipping the second fraction and multiplying.
For example, instead of looking at \[ \frac{\frac{p^{3}}{2q}}{-\frac{p^{2}}{4q}} \], you rewrite it as:

• \[ \frac{p^{3}}{2q} \times -\frac{4q}{p^{2}} \]

This keeps your calculations straightforward and helps avoid errors. Remember, flipping a fraction is the same as finding its reciprocal. This step is crucial to move forward with simplification.

Multiplication by Reciprocal

Multiplying by the reciprocal comes right after transforming the division into multiplication. A reciprocal of a number is simply one divided by that number. For fractions, flipping the numerator and the denominator gives you the reciprocal.
When we multiply by the reciprocal, like in

• \[ \frac{p^{3}}{2q} \times -\frac{4q}{p^{2}} \]

we should multiply the numerators together and the denominators together. Here, it results in:

• \[ -\frac{4p^{3}q}{2qp^{2}} \]

This step using reciprocals is why the division of fractions becomes multiplying. This simplification step not only makes calculations easier but also sets us up nicely for the cancellation of common factors.

Canceling Common Factors

Once multiplication is performed, the next important step is canceling out any common factors. Cancelling makes expressions simpler and is a vital aspect of fraction manipulation.
In the expression \[ -\frac{4p^{3}q}{2qp^{2}} \], both the numerator and the denominator have common factors: a \(q\) and a \(p^{2}\). When we cancel these out, we simplify the expression significantly:

• \[ -\frac{4p \cancel{p^2} \cancel{q}}{2 \cancel{q} \cancel{p^2}} \]

Canceling means you're dividing both the numerator and the denominator by the same value. This reduces complexity and makes the result easier to understand. This concept helps a lot in dealing with rational expressions, making them easier to solve in any algebraic operation.

Simplifying Rational Expressions

Simplification is the process of reducing an expression to its simplest form; for rational expressions, this often involves fractions. After canceling common factors in

• \[ -\frac{4p}{2} \]

, the expression can be further simplified by dividing the remaining terms. This gives us:

• \[ -2p \]

Simplifying rational expressions involves reducing them to a form where no further simplification is possible.
It helps create a more concise expression from original complex fractions. Practicing this helps develop a keen eye for identifying relationships and commonalities within expressions. Simplifying makes continued algebraic operations, whether adding, subtracting, or further multiplication, more streamlined and manageable.