Problem 14 - Algebra 2 and Trigonometry

Question

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{14 b^{4}}{21 b^{3}}\)

Step-by-Step Solution

Answer

The simplest form is \(\frac{2b}{3}\), undefined for \(b=0\).

Step 1: Identify Common Factors

Begin by identifying the common factors in the numerator and the denominator of the rational expression \(\frac{14b^4}{21b^3}\). Here, both 14 and 21 can be divided by their greatest common divisor, which is 7, and there is a common factor of \(b^3\) in both terms.

Step 2: Simplify the Coefficients

Divide the coefficients 14 and 21 by 7. With \(\frac{14}{7} = 2\) and \(\frac{21}{7} = 3\), the expression becomes \(\frac{2b^4}{3b^3}\).

Step 3: Simplify the Variable Terms

Simplify \(\frac{b^4}{b^3}\) by subtracting the exponents: \(b^{4-3} = b\). Thus, the expression simplifies to \(\frac{2b}{3}\).

Step 4: Identify Undefined Values

The original expression \(\frac{14b^4}{21b^3}\) is undefined when the denominator equals zero. Setting the denominator, \(21b^3 = 0\), gives \(b^3 = 0\), which means \(b = 0\).

Key Concepts

Greatest Common Divisor

The concept of the Greatest Common Divisor (GCD) is central to simplifying rational expressions. In this context, the GCD allows us to find the largest number that evenly divides both the numerator and the denominator of a fraction. By identifying this number, we can significantly simplify expressions.

In the expression \(\frac{14b^4}{21b^3}\), our task is to find the GCD of 14 and 21. We determine that this is 7 because 7 divides both number evenly:

• \(14 \div 7 = 2\)
• \(21 \div 7 = 3\)

By dividing both the numerator and the denominator by their GCD, we reduce the fraction to \(\frac{2b^4}{3b^3}\). This step simplifies the coefficients, making the entire rational expression simpler to handle in further steps.

Identifying Undefined Values

After simplifying an expression, it's crucial to identify the values that would make the expression undefined. A rational expression is undefined when the denominator equals zero since division by zero is not possible.

For the expression \(\frac{14b^4}{21b^3}\), the denominator is \(21b^3\). To find when this expression is undefined, set the denominator equal to zero:

• \(21b^3 = 0\)
• Solve for \(b\): \(b^3 = 0\), which implies \(b = 0\).

Thus, the rational expression is undefined when \(b = 0\). Being aware of undefined values is vital since it helps avoid errors in solving equations and understanding the domain of the expression.

Rational Expressions Simplification

Simplifying rational expressions helps in reducing them to their most manageable form. This involves canceling out common factors between the numerator and denominator, as well as manipulating variables appropriately.

Once we apply the GCD to the coefficients, resulting in \(\frac{2b^4}{3b^3}\), we then focus on the variables. The task here is to reduce \(b^4\) over \(b^3\) by subtracting their exponents, following the laws of exponents. This means:

• \(b^4\) divided by \(b^3\) equals \(b^{4-3}\)
• So the expression simplifies to \(b^1\) or just \(b\)

The outcome, \(\frac{2b}{3}\), presents the simplest form of the original expression. Simplifying helps in subsequent mathematical operations, making calculations smoother and more comprehensible. Always ensure to simplify expressions in both coefficients and variable terms for complete and efficient solutions.