Problem 29 - Algebra 2 and Trigonometry

Question

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ (3 b)^{2} \div \frac{3 b}{b+2} \cdot \frac{2 b+4}{b} $$

Step-by-Step Solution

Answer

The result is \(3(b+2)^2\) with \(b \neq -2\) and \(b \neq 0\).

Step 1: Simplify the Expressions in the Operation

First, identify and simplify any parts of the expression where possible. Notice that \( (3b)^2 \) simplifies to \( 9b^2 \). Similarly, the fraction \( \frac{2b+4}{b} \) can be rewritten by factoring the numerator: \( \frac{2(b+2)}{b} \).

Step 2: Convert the Division by a Fraction to Multiplication

Dividing by a fraction is equivalent to multiplying by its reciprocal. Convert \( \div \frac{3b}{b+2} \) into multiplication by flipping the fraction: \( \times \frac{b+2}{3b} \).

Step 3: Rewrite the Entire Expression Using Multiplication

The original expression \( (3b)^2 \div \frac{3b}{b+2} \cdot \frac{2b+4}{b} \) now becomes \( 9b^2 \times \frac{b+2}{3b} \times \frac{2(b+2)}{b} \).

Step 4: Simplify the Entire Expression

Perform multiplication across the fractions: \[9b^2 \cdot \frac{b+2}{3b} \cdot \frac{2(b+2)}{b} = \frac{9b^2(b+2)2(b+2)}{3b^2}\]Cancel out terms: - Cancel \(b^2\) in the numerator and denominator. - Cancel the 3 in \(9\) and the denominator: \( 9/3 = 3 \).The simplified expression is \( 3(b+2)^2 \).

Step 5: Identify Values That Make the Fractions Undefined

Fractions are undefined when the denominator equals zero. Identify all problematic values:- From \(\frac{3b}{b+2}\): \(b+2 eq 0\), so \(b eq -2\).- From \(\frac{2(b+2)}{b}\): \(b eq 0\).Therefore, \(b eq -2\) and \(b eq 0\).

Key Concepts

Simplifying Expressions

Simplifying expressions in mathematics involves reducing them to their simplest form without changing their value. It makes equations easier to solve or understand. In this particular exercise, we start with the expression \[(3b)^2 \div \frac{3b}{b+2} \cdot \frac{2b+4}{b}\]. The first step in simplifying is to look for ways to reduce individual components. For example:

• \( (3b)^2 \) simplifies to \( 9b^2 \) because multiplying a term by itself means multiplying the coefficients (3*3) and adding the exponents of like bases (b*b = b²).
• The fraction \( \frac{2b+4}{b} \) can be further simplified to \( \frac{2(b+2)}{b} \) by factoring out a common factor of 2 from the numerator.

Breaking expressions into smaller, more manageable parts can make the entire expression easier to handle.

Division of Fractions

Dividing by a fraction might seem complicated at first, but it becomes simple with a little trick. Instead of performing division directly, you can multiply by the reciprocal of the fraction. This means you flip the numerator and the denominator of the fraction.In our exercise, we have the division \( \div \frac{3b}{b+2} \). To handle this, we change it to multiplication by the reciprocal:

• \( \frac{3b}{b+2} \rightarrow \frac{b+2}{3b} \).

Once transformed, the original expression becomes \( 9b^2 \times \frac{b+2}{3b} \times \frac{2(b+2)}{b} \). This allows us to combine and simplify the fractions together, making the operation much more straightforward.

Undefined Values

Understanding undefined values is crucial when working with rational expressions. A fraction becomes undefined when its denominator is zero, as division by zero is not possible in mathematics.In this exercise, we determine the undefined values by setting each denominator in the fractions equal to zero and solving for the variable \(b\):

• From \( \frac{3b}{b+2} \), it's undefined for \( b+2 = 0 \) which gives \( b eq -2 \).
• From \( \frac{2(b+2)}{b} \), it's undefined for \( b = 0 \).

Thus, the rational expression is undefined for \(b = -2\) and \(b = 0\). It is critical to find these values to understand the limitations and restrictions of the solution.

Multiplication of Fractions

Multiplying fractions is a straightforward process involving the multiplication of numerators together and denominators together. We then simplify the resulting expression.After converting our expression to all multiplication: \[9b^2 \times \frac{b+2}{3b} \times \frac{2(b+2)}{b}\]We first multiply all numerators and denominators:

• The numerator becomes \(9b^2(b+2)2(b+2)\).
• The denominator becomes \(3b^2\).

By cancelling common terms (like \(b^2\)) and simplifying the fraction, we end up with:\[\frac{9b^2(b+2)2(b+2)}{3b^2} = 3(b+2)^2\]This shows how powerful simplification can be, allowing us to express complex fractions in a more manageable form.