Problem 24 - Algebra 2

Question

Simplify each expression. $$ \frac{3 x}{4 y^{2}}-\frac{y}{6 x} $$

Step-by-Step Solution

Answer

\( \frac{9x^2 - 2y^3}{12xy^2} \)

Step 1: Find the Common Denominator

To simplify the expression \( \frac{3x}{4y^2} - \frac{y}{6x} \), begin by finding a common denominator for the fractions. Both denominators are \( 4y^2 \) and \( 6x \). Their least common multiple (LCM) is \( 12xy^2 \).

Step 2: Adjust the Numerators

Adjust each fraction so they have the common denominator \( 12xy^2 \). For \( \frac{3x}{4y^2} \), multiply the numerator and denominator by \( 3x \) to get \( \frac{9x^2}{12xy^2} \).For \( \frac{y}{6x} \), multiply the numerator and denominator by \( 2y^2 \) to get \( \frac{2y^3}{12xy^2} \).

Step 3: Simplify the Expression

Subtract the adjusted fractions: \[ \frac{9x^2}{12xy^2} - \frac{2y^3}{12xy^2} = \frac{9x^2 - 2y^3}{12xy^2} \].This is the simplified form of the expression.

Key Concepts

Least Common Denominator

When dealing with rational expressions like \( \frac{3x}{4y^2} - \frac{y}{6x} \), a crucial first step is to find the least common denominator (LCD). This makes combining the fractions possible. The LCD is essentially the smallest expression that both denominators can divide into without leaving a remainder. Here, we have the denominators \(4y^2\) and \(6x\). To find the LCD:

• Identify the unique factors in each denominator. For \(4y^2\), the factors are \(4\) and \(y^2\). For \(6x\), it's \(6\) and \(x\).
• The least common multiple of the numeric coefficients \(4\) and \(6\) is \(12\).
• Combine the variable parts \(y^2\) and \(x\) to get \(xy^2\).

Putting it all together, the least common denominator is \(12xy^2\). This allows us to write both fractions over this common base, setting the stage for further simplification steps.

Numerator Adjustment

Once the least common denominator, \(12xy^2\), is identified, the next step is to adjust the numerators of the fractions. This means converting each fraction to have the same denominator by appropriately scaling the numerators. Here's how:

• For \(\frac{3x}{4y^2}\), multiply by \(\frac{3x}{3x}\), resulting in \(\frac{9x^2}{12xy^2}\).
• For \(\frac{y}{6x}\), multiply by \(\frac{2y^2}{2y^2}\), resulting in \(\frac{2y^3}{12xy^2}\).

This process ensures both fractions now have the same denominator, \(12xy^2\), and prepares them for direct subtraction. Adjusting numerators is a critical step that maintains the equality of the fraction while converting it to a common form.

Polynomial Subtraction

The last step in simplifying the expression \(\frac{3x}{4y^2} - \frac{y}{6x}\) involves subtracting the numerators once they share a common denominator. With the fractions expressed as \(\frac{9x^2}{12xy^2}\) and \(\frac{2y^3}{12xy^2}\), we can focus on polynomial subtraction:

• The denominators are already equal, \(12xy^2\).
• Subtract the numerators: \(9x^2 - 2y^3\).

The result is a single fraction, \(\frac{9x^2 - 2y^3}{12xy^2}\). This approach combines the fractions into a simpler form, concluding the process of simplification. Understanding polynomial subtraction in the context of common denominators empowers you to tackle more complex rational expressions effectively.