Problem 83 - Elementary Algebra

Question

For Problems \(75-92\), perform the indicated operations and express your answers in reduced form. $$ \frac{3}{2 x^{3}}+\frac{6}{x} $$

Step-by-Step Solution

Answer

The answer is \(\frac{3 + 12x^2}{2x^3}\).

Step 1: Identify the Least Common Denominator

The first step is to find the least common denominator (LCD) for the fractions involved. The denominators in this expression are \(2x^3\) and \(x\). The LCD for these two terms is \(2x^3\) because it is the smallest expression that both \(2x^3\) and \(x\) can divide into without leaving a remainder.

Step 2: Rewrite Each Fraction with the LCD

Next, rewrite each fraction so that they have the same denominator as the LCD. The first fraction, \(\frac{3}{2x^3}\), already has the correct denominator. For the second fraction \(\frac{6}{x}\), we must adjust the denominator to be \(2x^3\). To do this, multiply both the numerator and the denominator of the second fraction by \(2x^2\), which gives \(\frac{6 \times 2x^2}{x \times 2x^2} = \frac{12x^2}{2x^3}\). Now both fractions have the same denominator.

Step 3: Combine the Fractions

With the same denominator, the fractions can now be combined into a single fraction: \(\frac{3}{2x^3} + \frac{12x^2}{2x^3} = \frac{3 + 12x^2}{2x^3}\).

Step 4: Simplify the Expression

Lastly, simplify the combined fraction if possible. The expression \(\frac{3 + 12x^2}{2x^3}\) cannot be simplified further since there are no common factors in the numerator and the denominator besides 1. Thus, the reduced form of the expression remains \(\frac{3 + 12x^2}{2x^3}\).

Key Concepts

Least Common Denominator

When dealing with fractions, especially in algebra, the least common denominator (LCD) is a crucial step for adding or subtracting them. The LCD is the smallest expression that each denominator can divide into without a remainder. Here's how you can find it:

• Identify the denominators in the problem. In our exercise, they are \(2x^3\) and \(x\).
• Break each denominator into its prime factors. The factored form of \(2x^3\) is \(2 \times x \times x \times x\) and for \(x\) it is simply \(x\).
• Select each factor the greatest number of times it appears in any factorization. For \(2x^3\) and \(x\), the LCD is \(2x^3\), because this contains all factors needed to cover both denominators.

Knowing the LCD allows us to combine fractions seamlessly, as it gives them a common foundation.

Fraction Operations

Once you have identified the least common denominator, the next step in fraction operations is to rewrite each fraction using this common denominator. This process ensures you can easily add, subtract, or otherwise combine the fractions.Here's how this was done in our problem:

• The first fraction \(\frac{3}{2x^3}\) didn’t need any changes because it already had the LCD as its denominator.
• The second fraction \(\frac{6}{x}\) needed adjustment. To convert \(x\) into \(2x^3\), multiply the numerator and the denominator by \(2x^2\). This gives \(\frac{12x^2}{2x^3}\).

Now both fractions share the same denominator. With aligned denominators, you can add the numerators directly, resulting in a single fraction: \(\frac{3 + 12x^2}{2x^3}\). This method ensures that all parts of each fraction are properly accounted for, allowing seamless arithmetic operations.

Simplifying Expressions

The last core concept in handling fractions is simplifying the final expression. Simplification involves reducing fractions to their simplest form.For our exercise:- After combining the fractions, the result was \(\frac{3 + 12x^2}{2x^3}\).- To simplify, you need to look for common factors in the numerator and the denominator.However, in this case, the numerator \(3 + 12x^2\) and the denominator \(2x^3\) do not share common factors other than 1. Hence, the expression \(\frac{3 + 12x^2}{2x^3}\) is already in its simplest form.Simplifying expressions involves making sure they cannot be reduced further, which can help make mathematical problems easier to work with and understand.