Problem 74 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{2}{x^{2}}+\frac{5}{x^{2}} $$

Step-by-Step Solution

Answer

\( \frac{7}{x^2} \)

Step 1: Identify Common Denominator

Both terms in the expression \( \frac{2}{x^2} + \frac{5}{x^2} \) already have the same denominator, \( x^2 \). This means we can directly add the numerators over this common denominator.

Step 2: Combine Numerators

Since the denominators are identical, combine the numerators: \( 2 + 5 \), to form a single fraction: \( \frac{2 + 5}{x^2} \).

Step 3: Simplify the Expression

Add the numerators together: \( 2 + 5 = 7 \). Thus, the expression simplifies to \( \frac{7}{x^2} \).

Key Concepts

Fractions

Fractions represent parts of a whole and are written in the form \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. They are commonly used in algebra to express quantities that are not whole numbers.
A fraction can be thought of as dividing the numerator by the denominator. For example, \( \frac{2}{3} \) means "2 divided by 3."
When working with fractions, especially in algebra, it is important to understand:

• How to locate and use numerators and denominators effectively.
• The relationships between different fractions.

Understanding fractions is foundational because many algebraic operations involve adding, subtracting, multiplying, or dividing them.

Simplification

Simplification in algebra refers to making an expression as simple as possible, reducing it to its smallest form. This usually involves combining like terms, reducing fractions, or applying algebraic rules to make expressions easier to work with.
In the case of fractions, simplification can involve:

• Combining fractions with the same denominator by adding or subtracting their numerators.
• Reducing a fraction to its lowest terms by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it.

Simplification is crucial in solving algebraic expressions efficiently, as it turns complex expressions into more manageable ones.

Common Denominator

The common denominator is a shared multiple of the denominators of two or more fractions. Having a common denominator is essential when adding or subtracting fractions because it allows the numerators to be directly combined.
In our example, the fractions \( \frac{2}{x^2} \) and \( \frac{5}{x^2} \) already have a common denominator, \( x^2 \). This makes the process of combining these fractions straightforward:

• Identify if the fractions already share a common denominator.
• If they do, as in our example, add the numerators and express the result over the common denominator.

Finding a common denominator can sometimes involve calculating the least common multiple (LCM) of different denominators. However, when the denominators are already the same, the addition or subtraction process becomes much simpler.