Problem 58 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{8 x-64}{y}}{\frac{x^{2}-64}{y^{2}}} $$

Step-by-Step Solution

Answer

\( \frac{8y}{x + 8} \) is the simplified form.

Step 1: Identify the Complex Fraction

The given expression is \( \frac{\frac{8x - 64}{y}}{\frac{x^2 - 64}{y^2}} \). This is a complex fraction because it contains one fraction divided by another.

Step 2: Rewrite the Division as Multiplication

To simplify a complex fraction, we can rewrite it as a division of fractions: \( \frac{8x - 64}{y} \div \frac{x^2 - 64}{y^2} \). Division of fractions can be rewritten as multiplication by the reciprocal, so it becomes:\[\frac{8x - 64}{y} \times \frac{y^2}{x^2 - 64}\]

Step 3: Cancel Common Terms in the Expression

First, look for expressions that can be factored out or simplified. The term \( 8x - 64 \) can be factored as \( 8(x - 8) \), and the term \( x^2 - 64 \) is a difference of squares, which can be factored as \( (x - 8)(x + 8) \). Hence, we get:\[\frac{8(x - 8)}{y} \times \frac{y^2}{(x - 8)(x + 8)}\]

Step 4: Simplify by Canceling Common Factors

Now that we have the expression in a factored form, cancel any common factors present in the numerator and the denominator. Specifically, \( y \) in the numerator cancels with one \( y \) from \( y^2 \), and \( x - 8 \) cancels from the numerator and the denominator:\[8 \times \frac{y}{x + 8}\]

Step 5: Write the Final Simplified Expression

The final simplified form of the complex fraction is:\[\frac{8y}{x + 8}\]

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking a polynomial down into simpler polynomials that, when multiplied together, give back the original polynomial. This is a useful tool when simplifying algebraic expressions or solving equations.

In the given example, we encounter two polynomials that need factoring:

• Firstly, the expression \(8x - 64\). We factor out the greatest common factor, which is 8, giving us \(8(x - 8)\).
• Secondly, \(x^2 - 64\) is a difference of squares. This particular form, \(a^2 - b^2\), always factors into \((a - b)(a + b)\). So, \(x^2 - 64\) becomes \((x - 8)(x + 8)\).

Understanding how to factor these polynomials is crucial in simplifying the complex fraction. By breaking down the polynomials, we exposed common factors that we can cancel out in later steps, simplifying the expression further.

Multiplication of Fractions

Multiplying fractions might seem tricky, but it can be straight-forward with a few key ideas:

Instead of dividing by a fraction, \(\frac{a}{b} \div \frac{c}{d}\), we multiply by its reciprocal: \(\frac{a}{b} \times \frac{d}{c}\). This flips the second fraction, making it easier to process.

For our complex fraction, we used this principle:

• We converted \(\frac{8x - 64}{y} \div \frac{x^2 - 64}{y^2}\) into \(\frac{8x - 64}{y} \times \frac{y^2}{x^2 - 64}\).
• Multiplying becomes straightforward here. Simply multiply across the numerators and across the denominators.
• Our multiplication also allows us to later identify and cancel common factors, aiding in further simplification.

By shifting from division to multiplication using reciprocals, we've turned a complex fraction into an easier multiplication problem that quickly reveals common factors we can cancel.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form, making them easier to understand and work with.

In simplifying the expression from the given problem, we:

• Identified common factors between the polynomials involved \( (x - 8)\), in both the numerator and the denominator.
• Canceled out these factors, which reduced the expression further.
• The expression was thus streamlined to \(8 \times \frac{y}{x + 8}\), where no further reduction was possible.

Understanding when and how to cancel these common factors is crucial. It prevents errors and ensures you aren't performing unnecessary steps. Especially with complex fractions, simplifying algebraic expressions correctly brings clarity and ease to what initially seems a daunting task.