Problem 48 - Algebra for College Students

Question

Simplify each complex fraction. $$ \frac{\frac{9}{x}+\frac{7}{x^{2}}}{\frac{5}{y}+\frac{3}{y^{2}}} $$

Step-by-Step Solution

Answer

\( \frac{(9x + 7) y^2}{x^2(5y + 3)} \).

Step 1: Identify the Complex Fraction

The expression given is \( \frac{\frac{9}{x}+\frac{7}{x^{2}}}{\frac{5}{y}+\frac{3}{y^{2}}} \). This is a complex fraction, which means a fraction where the numerator and/or the denominator is also a fraction.

Step 2: Simplify the Numerator and Denominator

Simplify both the numerator \( \frac{9}{x} + \frac{7}{x^2} \) and the denominator \( \frac{5}{y} + \frac{3}{y^2} \). To do this, find a common denominator for each.

Step 3: Find Common Denominator for the Numerator

The common denominator for \( \frac{9}{x} \) and \( \frac{7}{x^2} \) is \( x^2 \). Rewrite the fractions: \( \frac{9x}{x^2} + \frac{7}{x^2} = \frac{9x + 7}{x^2} \).

Step 4: Find Common Denominator for the Denominator

The common denominator for \( \frac{5}{y} \) and \( \frac{3}{y^2} \) is \( y^2 \). Rewrite the fractions: \( \frac{5y}{y^2} + \frac{3}{y^2} = \frac{5y + 3}{y^2} \).

Step 5: Write the Division of Fractions

The complex fraction becomes \( \frac{\frac{9x + 7}{x^2}}{\frac{5y + 3}{y^2}} \), which is equivalent to \( \frac{9x + 7}{x^2} \div \frac{5y + 3}{y^2} \).

Step 6: Apply the Division Rule

To divide by a fraction, multiply by its reciprocal. So, \( \frac{9x + 7}{x^2} \times \frac{y^2}{5y + 3} \).

Step 7: Multiply the Fractions

Multiply the numerators and multiply the denominators: \( \frac{(9x + 7) \, y^2}{x^2(5y + 3)} \).

Step 8: Simplify if Possible

Check for any common factors between the numerator and the denominator. In this case, there are no common factors to simplify further.

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, we aim to express the fraction in a manner that is easier to understand, without changing its value. This involves reducing the fraction to its simplest form. To simplify fractions, follow these steps:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.
• Express the fraction using the results from the division.

In your exercise, the complex fraction involves both a numerator and a denominator made up of separate fractions. Simplifying involves finding a common denominator for each, which allows us to combine terms and reduce the overall complexity of the fraction. This process helps to simplify complex algebraic expressions efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. Dealing with algebraic expressions requires handling them like you would any numbers but with additional rules because of the variables.The components of algebraic expressions include:

• Variables, such as \(x\) and \(y\).
• Coefficients, which are numbers multiplying variables, like 9 in \(9x\).
• Operations such as addition, subtraction, multiplication, and division.

Algebraic expressions can be combined or separated by finding common denominators or expanding them, among other methods. In the given exercise, simplifying the algebraic expression within the complex fraction means working with expressions in the numerator and denominator separately before applying operations to the whole fraction.

Common Denominator

The concept of a common denominator is pivotal when you are dealing with adding or subtracting fractions. To simplify such expressions, each fraction needs to be rewritten with the same denominator.Here's how it works:

• Identify the common denominator, which is typically the least common multiple (LCM) of the existing denominators.
• Rewrite each fraction such that it shares this common denominator.
• This allows the numerators to be easily added or subtracted.

For example, with \( \frac{9}{x} + \frac{7}{x^2} \), the least common denominator is \(x^2\). Thus, we modify \( \frac{9}{x} \) to \( \frac{9x}{x^2} \) to share a common denominator with \( \frac{7}{x^2} \). Finding a common denominator facilitates combining algebraic expressions in the numerator and denominator in a complex fraction.

Multiplying Fractions

Multiplying fractions is one of the fundamental operations needed when simplifying complex fractions. This process involves a straightforward rule but requires careful attention.To multiply fractions:

• Multiply the numerators together to find the new numerator.
• Multiply the denominators together to find the new denominator.
• Simplify the result if possible, by dividing both the numerator and the denominator by any common factors.

For the complex fraction in the exercise, after rewriting the problem as a division of fractions, we multiply one fraction by the reciprocal of the other. This transforms \( \frac{9x + 7}{x^2} \div \frac{5y + 3}{y^2} \) into \( \frac{9x + 7}{x^2} \times \frac{y^2}{5y + 3} \). Multiplying these fractions results in a new fraction that represents the simplified form of the original complex fraction. In this specific case, no further simplification is needed as there are no common factors.