Problem 12 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{7 d^{2}}{6 d} \cdot \frac{12 d^{2}}{2 d}$$

Step-by-Step Solution

Answer

The product in simplest form is \(7d^{2}\)

Step 1: Multiply the numerators and the denominators

The first step is to multiply the numerators together and similarly the denominators together. This would give: \(\frac{7d^{2} \cdot 12d^{2}}{6d \cdot 2d}\)

Step 2: Simplify the obtained fractions

The multiplication gives: \(\frac{84d^{4}}{12d^{2}}\)

Step 3: Cancel out common factors

Factors that appear in both the numerator and the denominator can be cancelled out. Doing this, will result in: \(\frac{7d^{2}}{1}\)

Key Concepts

Simplifying Fractions

Simplifying fractions is about reducing a fraction to its simplest form. This means making sure that both the numerator and the denominator share no common factors other than 1, which results in a smaller or more "readable" expression.
To simplify a fraction, follow these steps:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.

Remember, if you multiply or divide both parts of a fraction by the same number (except zero), the value does not change.
In our example, after multiplying the fractions, we had: \[\frac{84d^4}{12d^2}\] Both the numerator and the denominator have the common factor of \(12d^2\), making the simplified form: \[\frac{7d^{2}}{1}, \text{ or just } 7d^2.\] By simplifying, we make calculations easier for future use.

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but it's straightforward when broken down into steps. When you multiply fractions, you multiply the numerators with one another and the denominators with one another. The resulting fraction is: \[\frac{numerator \times numerator}{denominator \times denominator}\] In our problem, we needed to multiply \(\frac{7d^{2}}{6d}\) by \(\frac{12d^{2}}{2d}\). Here’s how to do that:

• Multiply the numerators: \(7d^2 \times 12d^2 = 84d^4 \)
• Multiply the denominators: \(6d \times 2d = 12d^2 \)

The next step is to simplify the product, using our understanding from simplifying fractions.
Multiplying fraction numerators and denominators helps us combine different parts of rational expressions into a single fraction. This is crucial before simplifying further.

Polynomial Division

Polynomial division, particularly with fractions containing polynomials, is closely tied to simplifying expressions. Just like in regular division, you divide the numerator of the polynomial by the denominator. Here’s the process:

• If possible, factor both the numerator and the denominator of the polynomial.
• Cancel out any common factors between the numerator and the denominator.

In our example, the division occurred after multiplying the fractions. We got: \[\frac{84d^4}{12d^2}\] Here, both terms can be divided by \(12d^2\) (the GCF). This division simplifies to \(7d^2\), effectively reducing the complexity.
Polynomial division simplifies expressions so they can be more easily interpreted and utilized in further calculations or problems. It's an essential skill in algebra that aids in breaking down complex parts into manageable pieces.