Problem 5 - Beginning and Intermediate Algebra

Question

Find each product and simplify if possible. $$ -\frac{5 a^{2} b}{30 a^{2} b^{2}} \cdot b^{3} $$

Step-by-Step Solution

Answer

The simplified product is \(-\frac{b^2}{6}\).

Step 1: Simplify the Fraction

Let's begin by simplifying the initial fraction \(-\frac{5a^{2}b}{30a^{2}b^{2}}\). Since \(-5\) and \(30\) have a common factor of \(5\), divide each by \(5\) to get \(-\frac{1}{6}\). Next, cancel out the common \(a^{2}\) from both the numerator and the denominator to get \(-\frac{1\cdot b}{6\cdot b^{2}}\). Further, simplify this in terms of \(b\), which gives \(-\frac{1}{6b}\).

Step 2: Multiply by \(b^3\)

Now multiply the simplified fraction \(-\frac{1}{6b}\) by \(b^3\). This results in \(-\frac{b^3}{6b}\).

Step 3: Simplify the Expression

Further simplify \(-\frac{b^3}{6b}\) by cancelling one factor of \(b\) from the numerator and denominator to get \(-\frac{b^2}{6}\).

Key Concepts

Fraction Simplification

Before diving into multiplying fractions or dealing with algebra, it's important to start with simplifying each fraction to its simplest form. Fraction simplification is an essential skill in algebra.
It means reducing the fraction to its smallest numerator and denominator while maintaining the same value.
When you have a fraction such as \(-\frac{5a^{2}b}{30a^{2}b^{2}}\), the goal is to divide both the numerator and the denominator by their greatest common factor (GCF).
This makes the fraction easier to work with. In our example, both \(-5\) and \(30\) share the GCF of \(5\), which allows us to reduce it to \(-\frac{1}{6}\).
Next, we must simplify variables. The \(a^{2}\) in the numerator and the denominator cancels itself out, as they are the same, leaving \(-\frac{b}{6b^2}\).
Finally, further simplifying the \(b\) terms leads you to \(-\frac{1}{6b}\). Understanding how to effectively simplify fractions sets a solid foundation for algebraic operations.

Multiplying Fractions

Once you have your simplified fractions, it’s time to multiply them. This process is straightforward: multiply the numerators together and the denominators together.
For example, taking \(-\frac{1}{6b}\) and multiplying by \(b^{3}\) (which can be rewritten as \(\frac{b^3}{1}\)), results in \(-\frac{b^3}{6b}\).

• Numerator: Multiply \(-1\) and \(b^3\), which gives you \(-b^3\).
• Denominator: Multiply \(6b\) and \(1\), resulting in \(6b\).

This systematic approach, multiplying numerators and denominators separately, is the key to handling fractional multiplications.
This consistency makes it easy to manage even as expressions become more complex.

Cancelling Common Factors

After multiplying, the next step is to simplify the expression by canceling common factors. This means identifying terms in the numerator and denominator that are the same and removing them.
Let's look at \(-\frac{b^3}{6b}\), where \(b\) is a common factor. We can divide both the numerator and the denominator by \(b\), which simplifies \(-\frac{b^3}{6b}\) to \(-\frac{b^2}{6}\).

1. Identify common factors: Both \(b^3\) in the numerator and \(6b\) in the denominator have a \(b\) factor.
2. Divide by the common factor: Perform the division to cancel out a \(b\) from each.

This reduction process is crucial because it simplifies the expression and helps achieve the minimal form of an algebraic fraction, making further operations clearer and more manageable.
Effective simplification through canceling common factors is a powerful tool in algebraic manipulations.