Problem 7 - Algebra 1

Question

Simplify the expression. $$\frac{3}{10 x}-\frac{1}{4 x^{2}}$$

Step-by-Step Solution

Answer

\(\frac{6x - 5}{20x^2}\)

Step 1: Identify Like Terms

In the given expression \(\frac{3}{10 x}-\frac{1}{4 x^{2}}\), there are no like terms to combine directly. The denominators of the fractions are similar but not the same.

Step 2: Find Common Denominator

The fractions can be summed up if they have a common denominator. Identify the least common denominator (LCD) between \(10x\) and \(4x^2\). The LCD is the least common multiple of the two denominators which is \(20x^2\).

Step 3: Rewrite Fractions with Common Denominator

Now, rewrite each term with the common denominator. Recast \(\frac{3}{10x}\) and \(\frac{1}{4x^2}\) into equivalent fractions such that their denominator is \(20x^2\). By multiplying the numerator and the denominator of the first fraction by \(2x\), it becomes \(\frac{6x}{20x^2}\). The second fraction becomes \(\frac{5}{20x^2}\) when multiplied by 5.

Step 4: Subtract Fractions

Now that both fractions have the same denominator, they can be combined by subtracting the numerators: \(\frac{6x - 5}{20x^2}\)

Step 5: Simplify Resulting Fraction

The resulting fraction can't be simplified any further. So, \(\frac{6x - 5}{20x^2}\) is the simplified form of the given expression.

Key Concepts

Least Common Denominator

When simplifying expressions that involve fractions, one crucial task is finding the least common denominator (LCD). The LCD is the smallest multiple that is exactly divisible by each of the denominators in the fractions. This is necessary because fractions must have the same denominator if you want to perform addition or subtraction operations on them.

To find the LCD for the fractions \(\frac{3}{10x}\) and \(\frac{1}{4x^2}\), you need to determine the smallest multiple common to both 10x and 4x². You begin by comparing factors:

• The LCM of the numbers 10 and 4 is 20 since 20 is the smallest number that both 10 and 4 divide into without a remainder.
• The variable terms, x and x², have x² as the lowest common power since x² can accommodate x and still be shared with x².

Therefore, the least common denominator is \(20x^2\). Without it, you cannot easily subtract the fractions as intended.

Algebraic Expressions

Algebraic expressions use numbers, variables, and mathematical operations to represent value and relationships. In our expression \(\frac{3}{10x} - \frac{1}{4x^2}\), we deal with algebra by including variables (in this case, x) which represent quantities that can vary. Such expressions often require manipulation to simplify or solve within a given problem.

One important step in working with algebraic expressions is ensuring that you manage variables and coefficients together correctly. Variables can complicate finding the LCD and simplifying steps, as they don't behave as fixed numbers. Instead, think of them as placeholders or descriptors of numbers we don't readily clarify.

Manipulating expressions means we often involve:

• Rewriting terms using common factors or denominators for shared processes
• Performing operations with coefficients and maintaining 'like terms' (terms with the same variable parts and powers)

This logical structuring and manipulation of terms with variables are key in algebraic expressions.

Subtracting Fractions

Subtracting fractions, especially those with different denominators, requires careful aligning of the denominators through finding the least common denominator, as we previously discussed. Once each fraction is expressed with this common denominator, subtracting becomes straightforward.

For the expression \(\frac{3}{10x} - \frac{1}{4x^2}\), the revised expressions become \(\frac{6x}{20x^2}\) and \(\frac{5}{20x^2}\) respectively. To subtract these,

• Keep the common denominator \(20x^2\).
• Subtract the numerators: \(6x - 5\).

This results in the fraction \(\frac{6x - 5}{20x^2}\), which cannot be simplified further. Ensuring that fractions align through a common denominator is essential in maintaining balance and simplicity in subtraction.