Problem 18 - Elementary Algebra

Question

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{r^{2}+11 r+24}{9 r}}{\frac{r^{2}-64}{27 r^{3}}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3r^{2}(r+3)}{(r-8)} \).

Step 1 - Rewrite the Complete Expression

Start by rewriting the given complex fraction as a division problem: \[ \frac{\frac{r^{2}+11r+24}{9r}}{\frac{r^{2}-64}{27r^{3}}} = \frac{\frac{r^{2}+11r+24}{9r}}{\frac{r^{2}-64}{27r^{3}}} = \frac{r^{2}+11r+24}{9r} \times \frac{27r^{3}}{r^{2}-64}. \]

Step 2 - Factor the Polynomials

Factor the quadratic expressions in the numerator and the denominator. \[ r^{2}+11r+24 = (r+3)(r+8), \ r^{2}-64 = (r-8)(r+8). \] Substituting back, we get: \[ \frac{(r+3)(r+8)}{9r} \times \frac{27r^{3}}{(r-8)(r+8)}. \]

Step 3 - Cancel Common Terms

Look for common terms that can be cancelled in the numerator and denominator: \[ \frac{(r+3)(r+8)}{9r} \times \frac{27r^{3}}{(r-8)(r+8)} = \frac{(r+3)}{9} \times \frac{27r^{2}}{(r-8)} = \frac{(r+3) \times 27r^{2}}{9 \times (r-8)}. \]

Step 4 - Simplify the Remaining Expression

Perform the necessary multiplication and division: \[ \frac{(r+3) \times 27r^{2}}{9 \times (r-8)} = 3r^{2} \times \frac{(r+3)}{(r-8)} = \frac{27r^{2}(r+3)}{9(r-8)} = \frac{3r^{2}(r+3)}{(r-8)}. \]

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a complex structure into simpler building blocks. When you factor a polynomial, you express it as a product of its simpler factors.

For example, consider the quadratic polynomial \(r^{2} + 11r + 24\). It can be factored into two binomials: \((r + 3)(r + 8)\). Similarly, a difference of squares like \(r^{2} - 64\) can be factored into \((r - 8)(r + 8)\). Recognizing these patterns helps simplify expressions and solve equations more efficiently.

Multiplying Fractions

When multiplying fractions, you multiply the numerators together and the denominators together.

For the complex fraction \(\frac{\frac{r^{2}+11r+24}{9r}}{\frac{r^{2}-64}{27r^{3}}}\), we first rewrite it as a division problem, which involves multiplying by the reciprocal.

This turns into: \(\frac{r^{2}+11r+24}{9r} \times \frac{27r^{3}}{r^{2}-64}\). Now, multiply the individual numerators and denominators to get the complete multiplied fraction.

Canceling Common Factors

One of the most useful simplifications in algebra involves canceling common factors. After factoring the numerators and denominators, look for terms that are exactly the same and appear both in the numerator and the denominator.

In our example, factor the terms: \(\frac{(r+3)(r+8)}{9r} \times \frac{27r^{3}}{(r-8)(r+8)}\). The \((r+8)\) terms in the numerator and denominator cancel out, leaving you with: \(\frac{(r+3) \times 27r^{2}}{9 \times (r-8)}\). Recognizing and canceling these common factors simplifies the problem considerably.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operation symbols. Simplifying them often involves many steps, including factoring, multiplying, and canceling terms.

Consider \(\frac{(r+3) \times 27r^{2}}{9 \times (r-8)}\). Simplify each part by performing the operations: \(27r^{2}\) divided by \(9\) simplifies to \(3r^{2}\). So, the expression becomes: \(3r^{2} \times \frac{(r+3)}{(r-8)}\). The final simplified form is \(\frac{3r^{2}(r+3)}{(r-8)}\). Understanding these steps helps in dealing with any algebraic expression efficiently.