Problem 77 - Intermediate Algebra

Question

Perform the operations and simplify. $$ \frac{10 r^{2} s}{6 r s^{2}} \cdot \frac{3 r^{3}}{2 r s} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{5r^3}{2s^2} \).

Step 1: Multiply the Fractions

Multiply the numerators of both fractions together. Multiply the denominators of both fractions as well. The expression becomes: \( \frac{10r^2 s \cdot 3r^3}{6rs^2 \cdot 2rs} = \frac{30r^5 s}{12r^2 s^3} \).

Step 2: Simplify the Coefficients

Divide the coefficients (30 in the numerator and 12 in the denominator) by their greatest common divisor, which is 6. Therefore, \( \frac{30}{12} = \frac{5}{2} \). The fraction becomes: \( \frac{5r^5 s}{2r^2 s^3} \).

Step 3: Simplify the Variables

Simplify the powers of \( r \) and \( s \) by subtracting the exponents. For \( r \), subtract 2 from 5: \( r^{5-2} = r^3 \). For \( s \), subtract 3 from 1: \( s^{1-3} = s^{-2} \) or \( \frac{1}{s^2} \). The expression becomes \( \frac{5r^3}{2s^2} \).

Key Concepts

Rational Expressions

Rational expressions are similar to fractions but consist of polynomials in both the numerator and the denominator. Just like fractions, they can be manipulated by applying basic arithmetic operations of addition, subtraction, multiplication, and division. These expressions are "rational" because they represent ratios of polynomials. In our original exercise, \( \frac{10r^2 s}{6rs^2} \cdot \frac{3r^3}{2rs} \), both the numerator and denominator of each fraction are polynomial expressions. To simplify rational expressions effectively, it's crucial to understand how to handle polynomial terms, especially through factoring and reducing common terms much like you would simplify simple fractions.

Simplification of Fractions

The simplification of fractions involves reducing the fraction to its most basic form. This is done by dividing both the numerator and denominator by their greatest common divisor (GCD). In polynomial terms, this could mean factoring out common variables or coefficients. For the given rational expression, after multiplying, we have \( \frac{30r^5 s}{12r^2 s^3} \). By simplifying the coefficients first, dividing 30 and 12 by their GCD, which is 6, we simplify the fraction to \( \frac{5r^5 s}{2r^2 s^3} \). Once the coefficients are simplified, we can then simplify the variable parts by subtracting the exponents of like terms.

Multiplying Fractions

Multiplying fractions involves multiplying the numerators together to get the new numerator, and multiplying the denominators together to get the new denominator. This operation follows the same principle whether dealing with numbers or polynomial expressions. In our exercise, we multiplied \( \frac{10r^2 s}{6rs^2} \) by \( \frac{3r^3}{2rs} \), producing \( \frac{30r^5 s}{12r^2 s^3} \). Ensure to handle each part of the fractions carefully and simplify afterward to make the expression more manageable. Keeping track of all polynomial terms and simplifying them through factorization or combined operations is key to getting the correct simplified form."

Exponents and Powers

Exponents denote the number of times a base is multiplied by itself. Simplifying expressions with exponents, especially in rational expressions, requires understanding how to apply the rules of exponents. The rule

• \( a^m \/ a^n = a^{m-n} \)

helps us simplify expressions by subtracting the exponents of like bases. In the expression \( \frac{5r^5 s}{2r^2 s^3} \), for instance, we apply the rule to simplify \( r^{5-2} = r^3 \) and \( s^{1-3} = s^{-2} \). Negative exponents represent reciprocal bases, \( s^{-2} \) transforms to \( \frac{1}{s^2} \). Mastery of these principles allows the simplification of even complex algebraic expressions.