Problem 39 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( x y^2 \).

Step 1: Write the expression

The expression we need to simplify is \( \frac{x^{2}}{y} \cdot \frac{y^{3}}{x} \). This involves multiplication of two fractions.

Step 2: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together. This gives us: \[ \frac{x^2 \cdot y^3}{y \cdot x}. \]

Step 3: Simplify the Numerator

The numerator is \( x^2 \cdot y^3 \). There is no further simplification needed for the numerator itself.

Step 4: Simplify the Denominator

The denominator is \( y \cdot x \). There is no further simplification needed for the denominator itself.

Step 5: Cancel Common Factors

Look for common factors in the numerator and the denominator. The \( x \) and one \( y \) can be canceled out, leading to: \[ \frac{x \cdot y^2}{1}. \]

Step 6: Final Simplification

Since the denominator is 1, the expression simplifies further to \( x \, y^2 \).

Key Concepts

Exponent Rules

When dealing with fractions containing variables like in our expression, understanding exponent rules can be incredibly helpful. Exponents, or powers, indicate how many times a number, or variable, is multiplied by itself. For instance, \( x^2 \) means \( x \) is multiplied by itself twice, \( x \times x \). In multiplication scenarios like this, the power rule states that when you multiply two terms with the same base, you add their exponents.

• \( a^m \cdot a^n = a^{m+n} \)

In this exercise, our bases \( x \) and \( y \) are in the same expression but in different parts (numerator or denominator), so direct addition through multiplication doesn’t directly apply, but understanding these rules helps in simplifying larger problems.Remember, while exponent rules are key for terms with identical bases, they don't directly affect separate bases like in the given fractions, where simplification is more about canceling factors than altering exponents themselves.

Simplifying Fractions

Simplifying fractions involves reducing the terms to their simplest form where the greatest common divisor (GCD) of the numerator and denominator is 1.When you multiply fractions, start by multiplying the numerators together and then the denominators together. This initial multiplication gave:

• Numerator: \( x^2 \cdot y^3 \)
• Denominator: \( y \cdot x \)

The complex part is reducing this product to a simpler fraction. In this scenario, it's essential to look for opportunities to divide by common factors between the numerator and denominator to simplify. For instance, here, simplifying \( \frac{x^2 \cdot y^3}{y \cdot x} \) requires canceling out like terms that appear in both the numerator and the denominator. As you simplify, you ensure the expression remains equivalent in value, only easier to interpret or use further in calculations.

Canceling Common Factors

Canceling common factors is a handy trick in fraction multiplication or division that significantly simplifies expressions. Here’s the step-by-step for tackling this efficiently:Observe the full fraction \( \frac{x^2 \cdot y^3}{y \cdot x} \). Look for terms that appear both in the numerator and the denominator.

• The term \( x \) appears in both \( x^2 \) (numerator) and \( x \) (denominator).
• Similarly, \( y \) is present in both \( y^3 \) and \( y \).

By canceling one \( x \) from both numerator and denominator, you end up with \( x \cdot y^3 \) in the numerator and further simplify \( y^3 \) by canceling one \( y \) to end up with \( y^2 \).Canceling hence reduces the fraction to: \( \frac{x \cdot y^2}{1} \), ultimately giving \( x \cdot y^2 \), which simplifies effectively to just \( x y^2 \) as having a 1 in the denominator means it does not change the value. This process of canceling is vital to maintaining balance in the mathematical expression while simplifying your calculations.