Add or subtract, as indicated. See Example 4. (1/a)+(b/a²)

Verified step by step guidance

1. Identify the given expression: \(\frac{1}{b} + \frac{1}{a^{2}}\).
2. To add these two fractions, find a common denominator. The denominators are \(b\) and \(a^{2}\), so the common denominator is \(a^{2}b\).
3. Rewrite each fraction with the common denominator \(a^{2}b\): multiply numerator and denominator of \(\frac{1}{b}\) by \(a^{2}\) to get \(\frac{a^{2}}{a^{2}b}\), and multiply numerator and denominator of \(\frac{1}{a^{2}}\) by \(b\) to get \(\frac{b}{a^{2}b}\).
4. Now that both fractions have the same denominator, add the numerators: \(\frac{a^{2}}{a^{2}b} + \frac{b}{a^{2}b} = \frac{a^{2} + b}{a^{2}b}\).
5. The expression is now combined into a single fraction: \(\frac{a^{2} + b}{a^{2}b}\). This is the simplified sum of the original fractions.

Key Concepts

Adding and Subtracting Algebraic Fractions

To add or subtract algebraic fractions, first find a common denominator. This often involves factoring expressions to identify the least common denominator (LCD). Once the denominators match, combine the numerators accordingly and simplify the resulting expression.

Factoring and Simplifying Expressions

Factoring expressions like a² into a·a helps in identifying common denominators and simplifying fractions. Simplification involves reducing fractions by canceling common factors in the numerator and denominator to express the result in simplest form.

Understanding Variables and Exponents

Variables represent unknown values, and exponents indicate repeated multiplication. Recognizing that a² means a·a is crucial for manipulating algebraic fractions, especially when finding common denominators or simplifying terms.