Problem 33 - Prealgebra

Question

Multiply each of the following. Be sure all answers are written in lowest terms. $$\frac{a^{2} b}{c} \cdot \frac{c^{3}}{a b^{2}}$$

Step-by-Step Solution

Answer

The simplified product of the fractions is \( \frac{a c^2}{b} \).

Step 1: Write Down the Expression

The given expression is \( \frac{a^{2}b}{c} \cdot \frac{c^{3}}{ab^{2}} \). This is a multiplication of two fractions.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \( a^2 b \cdot c^3 \). This results in \( a^2 b c^3 \).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \( c \cdot ab^2 \). This results in \( cab^2 \).

Step 4: Write the Combined Fraction

Write the expression as a single fraction: \( \frac{a^2 b c^3}{c a b^2} \).

Step 5: Simplify the Fraction

Cancel out the common factors between the numerator and the denominator. The \( a \) on the top with \( a \) from \( a^2 \) on the bottom results in \( \frac{a}{a} = 1 \). Similarly, \( b \) from \( b^2 \) cancels with \( b \) from \( b \), resulting in \( \frac{b}{b} = 1 \). Finally, \( c \) from the numerator is canceled with \( c \) from the denominator. Thus, we get \( \frac{ac^2}{b} \).

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, we mean reducing them to their simplest form. In mathematics, it's often easier to work with expressions that are simple and short. This helps to perform calculations faster and with better accuracy. For example, in fraction multiplication, simplifying the expression means getting rid of unnecessary factors.

• Identify similar terms: Before performing operations, see if you can spot any terms that are similar or can be reduced.
• Combine like terms: In mathematical expressions, similar terms can be combined to make operations easier. In the multiply fractions problem, we simplified the expression by combining and reducing terms.
• Always aim for the lowest terms: Keeping your answers in the simplest form allows for quick checking and easy comparisons with other expressions.

By understanding the concept of simplifying, you can make math problems much more approachable and quick to solve.

Numerators and Denominators

In fractions, everything boils down to numerators and denominators. Understanding these components is crucial in performing operations like addition, subtraction, and multiplication. The numerator sits on top, while the denominator is below. They help in understanding the size and the fraction of the whole part that is being represented.

• Numerator: This part of the fraction tells you how many parts of the whole you have. In our exercise, the numerator began as \( a^2 bc^3 \).
• Denominator: This describes the total number of equal parts. Initially, the denominator was \( cab^2 \).

To multiply fractions, you simply multiply across the numerators to get a new numerator and across the denominators for a new denominator. Throughout our process, we maintained an understanding of each component to find and simplify our final answer.

Cancelling Common Factors

Cancelling common factors is a significant step in simplifying fractions. Imagine it as cutting down to the bare essentials of your expression, which will make your calculations easier and your answers simpler. In our exercise, canceling common factors between the numerator and denominator made our solution far cleaner.

• Locate common factors: Carefully inspect the numerator and denominator to find similar factors.
• Cancel out: Once you find the same factor on top and bottom, you can cancel them out as they essentially value to 1. This was evident when \( a \), \( b \), and \( c \) were cancelled out in our solution.
• Result: The simplification resulted in \( \frac{ac^2}{b} \), a much more straightforward expression to work with.

By practicing cancelling common factors, you'll not only simplify expressions efficiently but also minimize the risk of errors in more complex mathematical problems.