Problem 92 - Algebra for College Students

Question

Perform the indicated operations. $$ \frac{(a b)^{2}}{(a+b)^{2}} \cdot \frac{(a+b)^{3}}{(a b)^{3}} $$

Step-by-Step Solution

Answer

\( \frac{a+b}{ab} \)

Step 1: Simplify the First Fraction

Simplify the first fraction \( \frac{(ab)^2}{(a+b)^2} \) by recognizing that the numerator and the denominator are both squared terms.

Step 2: Simplify the Second Fraction

Simplify the second fraction \( \frac{(a+b)^3}{(ab)^3} \) by recognizing that the numerator and the denominator are both cubed terms.

Step 3: Combine the Fractions

Combine the simplified forms of the fractions. The new fraction is: \( \frac{(ab)^2}{(a+b)^2} \times \frac{(a+b)^3}{(ab)^3} \)

Step 4: Cancel Common Terms

Cancel common terms. \( \frac{(ab)^2 (a+b)^3}{(a+b)^2 (ab)^3} \) Here \( (a+b)^2 \) and \( (a+b)^2 \) cancel out, and \( (ab)^2 \) and \( (ab)^2 \) cancel out.

Step 5: Simplify the Expression

After canceling the common terms, what remains is: \( \frac{a+b}{ab} \). This is the simplest form.

Key Concepts

algebraic fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Examples of algebraic fractions include \(\frac{2x+3}{x-1}\) and \(\frac{a^2+b^2}{a-b}\). Understanding how to manipulate these fractions is essential in many algebraic operations.
These fractions can often be simplified by factoring polynomials or by canceling common factors from the numerator and the denominator.

canceling terms

Canceling terms is a crucial step in simplifying algebraic expressions. It involves removing common factors from the numerator and the denominator. For instance, in the expression \(\frac{(ab)^2 (a+b)^3}{(a+b)^2 (ab)^3}\), \( (a+b)^2 \) in the numerator can cancel out with \( (a+b)^2 \) in the denominator, leaving simplifying the fraction much easier.

Always ensure you only cancel terms that are common factors. Incorrectly canceling non-common terms can lead to incorrect results.

simplification process

The simplification process involves various steps to make an algebraic expression easier to work with. In the given exercise, we start by:

• Recognizing the forms of the numerator and denominator. For example, if terms are squared or cubed.
• Combining the fractions.
• Canceling out common terms.

After following these steps, the expression \(\frac{(ab)^2 (a+b)^3}{(a+b)^2 (ab)^3}\) simplifies to \(\frac{a+b}{ab}\) by canceling out the common terms.

exponent rules

Understanding exponent rules is vital when simplifying expressions involving exponents. The key rules used in the exercise include:

• \( (a^m)^n = a^{mn} \): This allows combining powers.
• \( a^m \times a^n = a^{m+n} \): This helps add powers when multiplying similar bases.

In the simplified fractions, recognizing how the exponents interact with each other allows for effective cancellation of terms.
These rules ensure accurate simplification of algebraic expressions with exponents.