Problem 41 - Prealgebra

Question

Multiply or divide as indicated. $$\frac{x^{2}}{y^{3}} \div \frac{x^{3}}{y^{2}}$$

Step-by-Step Solution

Answer

\( \frac{x^2}{y^3} \div \frac{x^3}{y^2} = \frac{1}{xy} \).

Step 1: Set Up the Problem

The exercise involves dividing two fractions: \( \frac{x^2}{y^3} \div \frac{x^3}{y^2} \). Dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Multiply by the Reciprocal

The reciprocal of \( \frac{x^3}{y^2} \) is \( \frac{y^2}{x^3} \). So rewrite the division as a multiplication: \( \frac{x^2}{y^3} \times \frac{y^2}{x^3} \).

Step 3: Multiply the Numerators and Denominators

Multiply the numerators and the denominators: \( \frac{x^2 \times y^2}{y^3 \times x^3} \). This gives us the expression \( \frac{x^2 y^2}{y^3 x^3} \).

Step 4: Simplify the Expression

Simplify by canceling out common factors in the numerator and the denominator. Divide \(x^2\) by \(x^3\), which gives \(\frac{1}{x}\), because \(x^3\) cancels with two \(x\)'s, leaving one \(x\) in the denominator. Divide \(y^2\) by \(y^3\), which gives \(\frac{1}{y}\), because \(y^3\) cancels with two \(y\)'s, leaving one \(y\) in the denominator. This results in \( \frac{1}{xy} \).

Key Concepts

Multiplying Reciprocals

When you divide any fraction, you actually multiply it by its reciprocal. A reciprocal is simply flipping the numerator and the denominator. This means that instead of thinking about division, you can convert everything to multiplication.

• Find the reciprocal by swapping the places of the numerator and the denominator.
• If you have the fraction \( \frac{x^3}{y^2} \), the reciprocal becomes \( \frac{y^2}{x^3} \).
• Use the reciprocal in the multiplication step instead of dividing.

By multiplying reciprocals, not only does it change the operation from division to multiplication (which is often easier to compute), but it sets a strong foundation for subsequent simplification steps. Remember, flipping a fraction is key to making division problems far more manageable.

Simplifying Expressions

After multiplying, simplification is crucial. Simplifying helps you reduce algebraic fractions to their simplest form, often making them easier to interpret or further manipulate. Simplification involves canceling out common factors in both the numerator and the denominator.

• Identify common terms in the numerator and the denominator. These are terms that appear in both locations.
• In the expression \( \frac{x^2 y^2}{y^3 x^3} \), you notice that both \(x\) and \(y\) terms are on the top and bottom.
• Cancel out these common terms by subtracting their exponents, while keeping the operation balanced.

For example, \(x^2\) divided by \(x^3\) leaves \(x\) in the denominator because \(3 - 2 = 1\). Similarly, \(y^2\) divided by \(y^3\) results in \(y\) in the denominator. This operation leaves you with \(\frac{1}{xy}\). Simplification not only makes expressions neat but also more meaningful.

Dividing Fractions

While dividing fractions might initially seem complex, it actually turns into a straightforward multiplication once you know how to handle reciprocals. Here’s how to do it:

• Recognize that you need to replace the division with multiplication using the reciprocal of the divisor.
• Transform the division problem \( \frac{x^2}{y^3} \div \frac{x^3}{y^2} \) into \( \frac{x^2}{y^3} \times \frac{y^2}{x^3} \).
• Follow through with the multiplication process—multiplying the numerators together and the denominators together.

This conversion makes dividing fractions less intimidating and more akin to basic arithmetic. The core principle is to transform the division of fractions into a multiplication problem, which you then solve using your knowledge of arithmetic and algebraic simplification techniques. By converting to multiplication, you maintain a consistent approach across similar exercises.