Problem 54 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{4 x-3}{3 x^{2} y}+\frac{2 x+1}{4 x y^{2}}$$

Step-by-Step Solution

Answer

Thus, the simplified form of the given expression is \(\frac{16 x y - 12 y + 6 x^{2} + 3 x}{12 x^{2} y^{2}}\).

Step 1: Find the Common Denominator

The common denominator of the fractions \(\frac{4 x-3}{3 x^{2} y}\) and \(\frac{2 x+1}{4 x y^{2}}\) is the least common multiple of their denominators. In this case, the least common multiple of \(3 x^{2} y\) and \(4 x y^{2}\) is \(12 x^{2} y^{2}\).

Step 2: Convert the Fractions to Have the Common Denominator

Next, convert the fractions to equivalents that have this denominator. Multiply the numerator and denominator of the first fraction by \(4 y\) and the second by \(3 x\). This gives the fractions \(\frac{(4 x-3) * 4 y }{(3 x^{2} y) * 4 y}\) and \(\frac{(2 x+1) * 3 x}{(4 x y^{2}) * 3 x}\), which simplifies to \(\frac{16 x y - 12 y}{12 x^{2} y^{2}}\) and \(\frac{6 x^{2} + 3 x}{12 x^{2} y^{2}}\).

Step 3: Add the Fractions

Now, add the converted fractions: \(\frac{16 x y - 12 y}{12 x^{2} y^{2}} + \frac{6 x^{2} + 3 x}{12 x^{2} y^{2}} = \frac{16 x y - 12 y + 6 x^{2} + 3 x}{12 x^{2} y^{2}}\).

Key Concepts

Least Common Multiple

To simplify rational expressions like algebraic fractions, finding the least common multiple (LCM) of the denominators is the first crucial step. The LCM is the smallest expression that is a multiple of each denominator.
To find the LCM of the denominators \(3x^2y\) and \(4xy^2\), you should:

• List the factors of each term: For \(3x^2y\), the factors are \(3\), \(x^2\), and \(y\). For \(4xy^2\), the factors are \(2^2\), \(x\), and \(y^2\).
• Identify the highest power of each factor: Here, \(x^2\), \(y^2\), and the numerical coefficient \(12\) (found by taking the greatest coefficient that divides both numbers \(3\) and \(4\) which gives \(12\)).

Multiplying these together, you get the LCM: \(12x^2y^2\). Having the LCM helps in rewriting fractions with a common denominator, which is the backbone of simplifying and adding fractions.

Algebraic Fractions

Algebraic fractions are like regular fractions, but they contain algebraic expressions in the numerator, the denominator, or both. This means both parts of the fraction can have variables like \(x\) and \(y\).
Understanding algebraic fractions is essential for manipulating expressions in algebra. For example, in our problem, the terms \(4x - 3\) and \(2x + 1\) are in the numerators, with variables mixing with constants.

• These are treated algebraically: meaning we use operations such as addition, subtraction, multiplication, and division with attention to like terms and factorization.
• Dealing with algebraic fractions often involves finding commonalities and sometimes factoring to simplify.

An excellent grasp of these concepts leads to effective simplification and addition of such expressions, laying the foundation for understanding more complex rational expressions.

Simplification of Fractions

The goal of simplification is to make the fraction as simple as possible without losing essential features of the expression. Simplifying involves reducing the fraction so that the numerator and the denominator have no common factors other than 1.
To simplify the fractions, rewrite each fraction so they have a common denominator, in this case, \(12x^2y^2\). This involves:

• Multiplying the first fraction, \(\frac{4x - 3}{3x^2y}\), by \(\frac{4y}{4y}\), which gives \(\frac{16xy - 12y}{12x^2y^2}\).
• Multiplying the second fraction, \(\frac{2x + 1}{4xy^2}\), by \(\frac{3x}{3x}\), resulting in \(\frac{6x^2 + 3x}{12x^2y^2}\).

Simplifying each like this ensures that when you combine them, it's straightforward as all parts align neatly, facilitating addition or subtraction.

Common Denominator

The common denominator is an essential part of simplifying and adding or subtracting fractions. Having a common denominator allows you to combine the fractions, aligning the division part, and then simplifying the result.
Once you determine the least common multiple, apply this LCM to transform each fraction so both share this denominator. In this exercise:

• The fractions \(\frac{16xy - 12y}{12x^2y^2}\) and \(\frac{6x^2 + 3x}{12x^2y^2}\) have been rewritten with \(12x^2y^2\) as the common denominator. Once converted, the numerators are adjusted accordingly.
• Add the numerators directly: \(16xy - 12y + 6x^2 + 3x\).
• This combined numerator over the common denominator \(12x^2y^2\) gives you a unified expression that can then be simplified further if possible.

Understanding common denominators helps you break down complex algebraic fractions into manageable parts.