Problem 40 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{2 a^{2}-2 b^{2}}{2 a^{2}+2 b^{2}}$$

Step-by-Step Solution

Answer

The reduced form is \(\frac{a^2 - b^2}{a^2 + b^2}\).

Step 1 - Factor the Numerator

The numerator is given as \(2a^2 - 2b^2\). Notice that you can factor out a common factor of 2: \(2(a^2 - b^2)\).

Step 2 - Factor the Denominator

The denominator is given as \(2a^2 + 2b^2\). Similarly, factor out a common factor of 2: \(2(a^2 + b^2)\).

Step 3 - Simplify the Expression

Now you have the expression \(\frac{2(a^2 - b^2)}{2(a^2 + b^2)}\). The common factors of 2 in the numerator and denominator can be canceled out, resulting in \(\frac{a^2 - b^2}{a^2 + b^2}\).

Step 4 - Verify the Simplification

Ensure there are no further common factors between the numerator and the denominator. Since \(a^2 - b^2\) and \(a^2 + b^2\) have no common factors, the expression is in its simplest form.

Key Concepts

Factoring

When simplifying rational expressions, factoring is a key tool. Factoring involves breaking down a complicated expression into simpler parts, or 'factors', that multiply together to form the original expression. For instance, in this problem, the numerator is given as \(2a^2 - 2b^2\). You can factor out the common factor of 2 to get \(2(a^2 - b^2)\). Similarly, the denominator \(2a^2 + 2b^2\) also has a common factor of 2, giving \(2(a^2 + b^2)\). Simplifying the factors is crucial because it makes it easier to see and cancel out common elements between the numerator and denominator.

Common Factors

Identifying common factors between the numerator and the denominator simplifies rational expressions significantly. In this problem, 2 is a common factor in both the numerator and the denominator. By factoring out the 2, you simplify the rational expression to \(\frac{2(a^2 - b^2)}{2(a^2 + b^2)}\). You can then cancel out the common factor of 2, resulting in \(\frac{a^2 - b^2}{a^2 + b^2}\). When there are no further common factors left between the numerator and the denominator, the expression is in its simplest form.

Numerator and Denominator

The numerator and denominator are the top and bottom parts of a fraction, respectively. For a rational expression, simplifying involves making these as simple as possible while retaining their relationship. In this exercise, our numerator is \(2a^2 - 2b^2\) and the denominator is \(2a^2 + 2b^2\). After factoring out the common 2 from both, we get \(\frac{a^2 - b^2}{a^2 + b^2}\). No further common factors exist, meaning this fraction is now in its simplest form. Ensure that you always check for any common factors between the numerator and the denominator before declaring the expression fully simplified.