Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(\frac{4 a^{2} b}{a^{3} b^{2}}\right)\left(\frac{5 a^{2} b}{2 b^{4}}\right) $$

Step-by-Step Solution

Answer

\( \frac{10a}{b^4} \)

Step 1: Simplify Each Fraction

Begin by simplifying each fraction separately. For \( \frac{4 a^2 b}{a^3 b^2} \), you can cancel out common terms in the numerator and denominator. Divide both the numerator and the denominator by \( a^2 \), which gives \( \frac{4b}{ab^2} \). Then divide by \( b \), yielding \( \frac{4}{ab} \).For \( \frac{5 a^2 b}{2 b^4} \), again cancel common terms. Divide by \( b \) to get \( \frac{5 a^2}{2b^3} \).

Step 2: Multiply the Simplified Fractions

Now, multiply the simplified fractions. Multiply the numerators: \( 4 \cdot 5 a^2 \) resulting in \( 20 a^2 \). Multiply the denominators: \( a b \cdot 2 b^3 \) which gives \( 2 a b^4 \). Thus, the expression becomes \( \frac{20 a^2}{2 a b^4} \).

Step 3: Further Simplify the Product

Simplify the fraction \( \frac{20 a^2}{2 a b^4} \). Divide the numerator and denominator by \( 2a \). This results in \( \frac{10 a}{b^4} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is a key concept in algebra that involves reducing fractions to their simplest form. It means finding an equivalent fraction where the numerator and the denominator have no common factors other than 1.
To simplify a fraction, we first look for common factors shared by the numerator and denominator. Then, we divide both the top and bottom of the fraction by these common factors.
For example, in the algebraic expression \( \frac{4a^2b}{a^3b^2} \), we identify \( a^2 \) and \( b \) as common factors in both the numerator and denominator.

• Dividing \( 4a^2b \) by \( a^2 \) and \( b \), we get \( \frac{4}{ab} \).
• We perform similar steps for \( \frac{5a^2b}{2b^4} \), simplifying to \( \frac{5a^2}{2b^3} \).

By using this approach, we ensure that our fractions are easy to work with, particularly in complex algebraic expressions.

Multiplying Fractions

The process of multiplying fractions involves a straightforward technique: multiply the numerators to get a new numerator and multiply the denominators to get a new denominator.
This method extends directly to algebraic fractions or rational expressions. After simplifying, each fraction individually, we need to apply this multiplication principle.
For instance, from our task, once simplified fractions \( \frac{4}{ab} \) and \( \frac{5a^2}{2b^3} \), we multiply them:

• Numerators: \( 4 \times 5a^2 = 20a^2 \)
• Denominators: \( ab \times 2b^3 = 2ab^4 \)

This results in the new faction \( \frac{20a^2}{2ab^4} \).
This example shows that multiplying algebraic fractions requires careful handling of both numbers and variables.

Rational Expressions

Rational expressions are fractions wherein the numerator and the denominator are polynomials. To simplify or multiply rational expressions, the same principles apply as with numerical fractions.
It’s vital to remember that while simplifying, we aim to reduce complexities by canceling common factors just as we do with numbers.
This step-by-step approach aids in managing variables and exponents involved in expressions like \( \frac{4a^2b}{a^3b^2} \) and \( \frac{5a^2b}{2b^4} \).

• Each term with an identical base can be simplified separately by keeping the highest power from the exponents.
• Upon multiplying, ensure that no factor remains in common in the numerator or denominator to achieve a simplified form.

Thus, rational expressions extend the concept of fractions into the realm of algebra involving variables, making them crucial in advanced mathematics.