Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Write the rational expression in simplest form.\(\frac{24 y^{3}}{56 y^{7}}\)

Step-by-Step Solution

Answer

The simplified form of \(\frac{24 y^{3}}{56 y^{7}}\) is \(\frac{3}{7y^{4}}\).

Step 1: Factoring the Numerator and Denominator

Firstly, the numerator and denominator of the fraction need to be factored out. Thus, \(\frac{24 y^{3}}{56 y^{7}}\) can be factored out as \(\frac{2^3*3*y^{3}}{2^3*7*y^{7}}\).

Step 2: Cancel Out the Common Terms

From here, we are able to cancel out the common terms. We are left with \(\frac{3}{7*y^{4}}\). The power difference between \(y^7\) in the denominator and \(y^3\) in the numerator becomes \(y^4\) in the simplified expression.

Step 3: Final Simplified Form

After cancelling out the common terms, the simplified rational expression is \(\frac{3}{7y^{4}}\). This is the simplest form of the original rational expression.

Key Concepts

Factoring Polynomials

When working with rational expressions, one of the first and most crucial steps is factoring polynomials found in both the numerator and the denominator. Factoring involves breaking down a number or expression into its constituent elements, known as factors. For instance, in the original expression \(\frac{24y^{3}}{56y^{7}}\), we start by recognizing that 24 and 56 can be divided into smaller factors.

• 24 can be factored into \(2^3 \times 3\).
• 56 can be factored into \(2^3 \times 7\).
• For the polynomial parts, \(y^3\) and \(y^7\) are the powers of \(y\).

These factorizations help in identifying and canceling common terms between the numerator and the denominator, leading to simpler expressions. Understanding how to factor efficiently is key to simplifying more complex rational expressions.

Exponents and Powers

Exponents and powers are fundamental concepts in mathematics that help us express repeated multiplication concisely. In the expression \(\frac{24y^{3}}{56y^{7}}\), the exponents help track the number of times the base, \(y\), is multiplied by itself. For instance, \(y^{3}\) means \(y\times y\times y\).

When simplifying rational expressions with terms that have exponents, such as \(y^3\) and \(y^7\), we use the laws of exponents. One of the key rules is that when dividing like bases, you subtract the exponents: \(a^m/a^n = a^{m-n}\). Thus, \(y^7\) divided by \(y^3\) becomes \(y^{7-3} = y^4\).

This process shows how understanding the laws of exponents allows us to simplify expressions by reducing the powers and making calculations more manageable.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form so that they are easier to understand and use. In the context of rational expressions like \(\frac{24y^{3}}{56y^{7}}\), simplification happens after factoring out common bases in both the numerator and the denominator.

• After factoring as \(\frac{2^3\times 3\times y^3}{2^3\times 7\times y^7}\), we can cancel out \(2^3\) and common powers of \(y\).
• This results in \(\frac{3}{7y^{4}}\), where neither the numerator nor the denominator can be reduced further.

Fraction simplification helps reveal the essence of the expression, ensuring that it’s as straightforward as possible. This skill is important not just in algebra but in everyday calculations and problem-solving. By removing complexity, we are left with a clean, easy-to-read expression that retains the same numerical value or algebraic meaning.