Problem 29 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{(y-6)^{7}}{y-6} $$

Step-by-Step Solution

Answer

Question: Simplify the rational expression: \(\frac{(y-6)^7}{y-6}\) Answer: \((y-6)^6\)

Step 1: Identify the Greatest Common Factors (GCF) between the numerator and denominator

The numerator is \((y-6)^7\) and the denominator is \(y-6\). The GCF between the numerator and the denominator is \(y-6\).

Step 2: Divide both the numerator and the denominator by the GCF

Divide both the numerator and the denominator by the GCF, which is \(y-6\): $$ \frac{(y-6)^7}{y-6} \div \frac{y-6}{y-6} $$

Step 3: Simplify the expression

Cancel out the common terms in the numerator and the denominator: $$ \frac{(y-6)^7}{y-6} \div \frac{y-6}{y-6} = \frac{(y-6)^{7 - 1}}{1} $$

Step 4: Final Answer

The simplified rational expression in the lowest terms is: $$ (y-6)^6 $$

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is essential when simplifying rational expressions. It refers to the largest factor that divides two or more numbers or terms. In algebra, identifying the GCF in expressions helps in reducing fractions to their simplest form. For instance, to simplify the rational expression \(\frac{(y-6)^{7}}{y-6}\), we first identify \(y-6\) as the GCF between the numerator and the denominator since \(y-6\) is a factor common to both terms.

Simplifying involves dividing both the numerator and the denominator by this GCF, thereby reducing the complex expression into a more manageable form. The division step is crucial because it applies the fundamental property that a number divided by itself equals one, which in algebraic terms helps in 'canceling out' the common factors.

Algebraic Fractions

Algebraic fractions are fractions with polynomials in the numerator, the denominator, or both. Simplifying these fractions often involves finding the GCF of the terms as we would with numerical fractions. However, with algebraic fractions, instead of working with just numbers, we work with variable expressions. The key to simplifying an algebraic fraction is to factor both the numerator and the denominator and then reduce any common factors.

For example, with the algebraic fraction \(\frac{(y-6)^{7}}{y-6}\), we simplify by reducing common factors to achieve the lowest terms. This process not only makes the expression simpler but also aids in further algebraic manipulations such as addition, subtraction, multiplication, and division with other algebraic fractions.

Polynomial Division

Polynomial division is akin to long division with numbers, but involves dividing polynomials. It's a method used to simplify expressions and solve algebraic equations. In the context of our example \(\frac{(y-6)^{7}}{y-6}\), the division process shows that when we divide \(y-6\) out of \(y-6\) raised to the 7th power, we are left with \(y-6\) raised to the 6th power, as repeated factors are being reduced.Understanding the multi-step process
During polynomial division, if the same term is present in both the numerator and the denominator, as shown in our example, we can simplify the expression by subtracting the exponent in the denominator from that in the numerator. This is a shortcut that stems from the laws of exponents. When applied correctly, polynomial division greatly simplifies complex-looking expressions into more understandable and workable forms, enabling students to tackle more challenging algebraic problems with confidence.