Problem 58 - Prealgebra and Introductory Algebra

Question

Use the product \(\frac{x^{a}}{y^{b}} \cdot \frac{y^{t}}{x^{d}},\) where \(a, b, c,\) and \(d\) are all positive integers. If \(a>d\) and \(b>c,\) which variable appears in the denominator of the simplified product?

Step-by-Step Solution

Answer

The variable that appears in the denominator of the simplified product is \(y\).

Step 1: Write down the given product

The given product is \(\frac{x^{a}}{y^{b}} \cdot \frac{y^{c}}{x^{d}}.\)

Step 2: Apply the multiplication rule of fractions

We multiply the fractions, which gives \(\frac{x^{a} \cdot y^{c}}{y^{b} \cdot x^{d}}.\)

Step 3: Apply the rule of exponents for division

When dividing two powers that have the same base, we subtract the exponents. Doing this gives \(\frac{y^{c-b} \cdot x^{a-d}}{1}.\)

Step 4: Take given conditions into account

Since the conditions are \(a > d\) and \(b > c,\) we have \(x^{a-d}\) in the numerator and \(y^{b-c}\) in the denominator.

Step 5: Determine which variable appears in the denominator

From previous step, it is clear that y appears in the denominator of the simplified product.

Key Concepts

Multiplication Rule of Fractions

Understanding the multiplication rule of fractions is essential for simplifying algebraic expressions involving fractions. When multiplying two fractions, you simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For instance, consider \(\frac{a}{b} \cdot \frac{c}{d}\). To multiply these fractions, you take \(a \cdot c\) as the new numerator and \(b \cdot d\) as the new denominator, resulting in \(\frac{a \cdot c}{b \cdot d}\).

This operation is pivotal in algebra, especially when dealing with variables and exponents. When you encounter algebraic fractions like \(\frac{x^{a}}{y^{b}} \cdot \frac{y^{c}}{x^{d}}\), you apply the same principle: multiply the x-terms together and the y-terms together, yielding \(\frac{x^{a} \cdot y^{c}}{y^{b} \cdot x^{d}}\).

Rule of Exponents for Division

The rule of exponents for division streamlines the process of simplifying expressions with like bases. When you divide two expressions that have the same base, you subtract the exponent of the denominator from the exponent of the numerator. This is mathematically presented as \(x^{m} \div x^{n} = x^{m-n}\), where \(x\) is any base and \(m\) and \(n\) are exponents.

In our exercise, we have both \(x\) and \(y\) with exponents in the numerator and denominator. By applying the division rule, \(x^{a} \div x^{d} = x^{a-d}\) and \(y^{c} \div y^{b} = y^{c-b}\); however, remember to switch the subtraction order if \(b > c\) to comply with mathematical conventions, resulting in \(y^{b-c}\) in the denominator.

Simplifying Expressions with Variables

Simplifying expressions with variables often includes combining like terms, factoring, and applying the rules of exponents appropriately. In algebra, simplification aims to reduce an expression to its most basic form without changing its value.

When simplifying the product \(\frac{x^{a}}{y^{b}} \cdot \frac{y^{c}}{x^{d}}\), we must pay close attention to the variables' exponents. Given that both \(x\) and \(y\) appear in both the numerator and denominator, we simplify by subtracting the exponents of like bases following the previously mentioned rule of exponents for division. This simplification might lead to either positive or negative exponents, indicating the position of the base (numerator or denominator) in the final expression.

Positive Integers in Algebra

In algebra, positive integers often represent constants, coefficients, or exponents. It's significant to note that the rules we apply in simplifying expressions work consistently regardless of whether these integers are positive, negative, or even zero (with the exception that we cannot have zero as an exponent of a variable).

For our example, we specifically look at cases where \(a>d\) and \(b>c\), all being positive integers. Their positivity ensures that when we subtract the smaller exponent from the larger one, as directed by the exponents for division rule, we will always end up with a positive exponent. This subtractive operation indicates that the variable with the larger initial exponent will be found in the respective part of the fraction where the larger exponent was—numerator for \(a-d\) since \(a>d\), and denominator for \(b-c\) since \(b>c\).