Problem 9 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{12 a^{2} b^{5}}{-54 a^{2} b^{3}}$$

Step-by-Step Solution

Answer

The simplified fraction is \(-\frac{2b^2}{9}\).

Step 1: Identify the Common Factors

The numerator of the fraction is \(12 a^2 b^5\) and the denominator is \(-54 a^2 b^3\). First, identify the common factors in both the numerator and the denominator. Notice that \(a^2\) is common in both, and the powers of \(b\) can be simplified by the lowest power, which is \(b^3\).

Step 2: Factor Out Common Numerical Coefficient

Determine the numerical coefficient and find their greatest common factor (GCF). For 12 and 54, the GCF is 6.

Step 3: Simplify the Coefficients

Divide the numerical coefficients by their GCF: \(\frac{12}{6} = 2\) and \(\frac{-54}{6} = -9\).

Step 4: Simplify the Power of "a"

Since \(a^2\) is present in both the numerator and denominator, it can be completely cancelled out, simplifying the terms involving \(a\) to 1.

Step 5: Simplify the Power of "b"

For the variable \(b\), subtract the exponent in the denominator from the exponent in the numerator: \(b^{5-3} = b^{2}\).

Step 6: Write the Simplified Fraction

Combine all simplified parts: The fraction reduces to \(-\frac{2b^2}{9}\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is a crucial concept when simplifying algebraic fractions. It is the largest factor that divides two numbers or terms without leaving a remainder. In this exercise, to simplify the fraction \(\frac{12 a^{2} b^{5}}{-54 a^{2} b^{3}}\), we start by identifying the GCF of the numerical coefficients, 12 and 54.

• The factors of 12 are: 1, 2, 3, 4, 6, and 12.
• The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, and 54.

The largest common factor here is 6. By dividing both the numerator and the denominator by their GCF, 6, we simplify these parts of the fraction. This yields a simpler expression, making it easier to handle any remaining algebraic simplifications.

Cancelled Terms

Cancelled terms play a pivotal role in the simplification of algebraic fractions. In our example, once we have the expression \(\frac{2 a^{2} b^{5}}{-9 a^{2} b^{3}}\) after dividing by the GCF, it's key to notice the matching terms in the numerator and denominator:

• The term \(a^2\) appears in both parts, allowing them to be completely removed or 'cancelled'. Once cancelled, they reduce to 1, as any term divided by itself is 1.

This cancellation simplifies the fraction significantly, leaving us with only terms of \(b\) and the numerical coefficients. Such reductions are common in algebraic simplifications and make the operation much more straightforward, reducing complex expressions to simpler forms.

Exponent Rules

In algebra, exponent rules are essential for manipulating expressions with powers effectively, especially when simplifying fractions. For the algebraic fraction \(\frac{2 b^{5}}{-9 b^{3}}\), we apply these rules to simplify the powers of \(b\). Exponent rules state that when dividing like bases, you subtract the exponents:

• For \(b\), we subtract the denominator's exponent 3 from the numerator's exponent 5, yielding \(b^{5 - 3} = b^2\).

This results in a significantly simplified form of the fraction: \(-\frac{2b^2}{9}\). Understanding these rules is fundamental when working with powers, allowing students to manipulate and simplify expressions correctly.