Problem 15 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{5 a^{2} b^{2}}{11 a b} \cdot \frac{22 a^{3}}{15 a b^{2}}\)

Step-by-Step Solution

Answer

The answer is \(\frac{2a^3}{3b}\).

Step 1: Multiply the Numerators

First, multiply the numerators together: \(5a^2b^2\) and \(22a^3\). This gives us: \(5a^2b^2 \times 22a^3 = 110a^{5}b^2\).

Step 2: Multiply the Denominators

Next, multiply the denominators together: \(11ab\) and \(15ab^2\). This results in: \(11ab \times 15ab^2 = 165a^2b^3\).

Step 3: Form the Combined Rational Expression

Write the combined rational expression from the results of Step 1 and Step 2: \(\frac{110a^5b^2}{165a^2b^3}\).

Step 4: Cancel Common Factors

Find common factors in the numerator and the denominator. In this case, \(a^2\) and \(b^2\) are common in both the numerator and the denominator. When we cancel, we get:\[\frac{110a^{3}}{165b^{1}}\]

Step 5: Simplify the Coefficients

Now simplify the coefficients \(\frac{110}{165}\). The greatest common divisor (GCD) of 110 and 165 is 55. Divide both by 55:\[\frac{110}{55} = 2, \quad \frac{165}{55} = 3\]This gives us: \(\frac{2a^3}{3b}\).

Step 6: Final Expression in Simplest Form

After performing the indicated operations and simplifications, the final expression is \(\frac{2a^3}{3b}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra, as it helps make mathematical statements more easily manageable. When simplifying, our goal is to reduce expressions to their simplest form while preserving their original value. This ensures calculations are more straightforward and less prone to error.When dealing with expressions involving fractions, this process often involves finding common factors. Much like simplifying numerical fractions, simplifying algebraic expressions requires us to:

• Identify any common factors in the numerator and the denominator.
• Cancel out these common factors to reduce the expression.
• Ensure the remaining parts of the expression are as simple as possible.

For example, in a rational expression, the greatest common factor may exist among the coefficients, or the algebraic variables, such as terms like \(a^2\) or \(b\). By factoring and reducing, we streamline the solving process, enabling clearer insights into how variables interact.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions but involve variables along with numbers in the numerator and denominator. Just as with numerical fractions, understanding how to handle algebraic fractions is crucial for problem-solving in algebra. For algebraic fractions, the operations we can perform include addition, subtraction, multiplication, and division – all follow the basic rules established for fractions:

• **Numerator and Denominator:** The expressions on top and bottom function as typical fractions.
• **Common Factors:** Often, algebraic fractions require simplification, involving division by the greatest common factor.
• **Expression Complexity:** Various algebraic operations may demand initial simplification to reduce complexity.

When you multiply or divide these complex fractions, being acquainted with the properties of fractions allows clearer navigation through simplification and resolution. This is typically achieved by reducing expressions or adjusting terms to cancel effectively.

Multiplying Fractions

Multiplying fractions is a basic arithmetic operation that can also be applied to algebraic fractions, requiring a simple extension of the rules used for simple numerical fractions.In multiplication:

• **Multiply Numerators:** Involves combining the expressions in the numerators through multiplication.
• **Multiply Denominators:** Similarly, multiply the expressions in the denominators together.
• **Simplification:** The resulting expression often needs simplification, achieved by canceling out common factors between the new numerator and denominator.

For the given problem, multiplying two rational expressions required combining the numerators \(5a^2b^2\) and \(22a^3\), as well as the denominators \(11ab\) and \(15ab^2\). To simplify, we identified and canceled common factors, ultimately arriving at the simplest result of \(\frac{2a^3}{3b}\). This effective approach mirrors fundamental arithmetic principles while introducing algebraic complexity.