Problem 80 - Algebra for College Students

Question

Express each of the following as a single fraction involving positive exponents only. \(a^{-1}+a^{-1} b^{-3}\)

Step-by-Step Solution

Answer

\(\frac{b^3}{a(b^3 + 1)}\)

Step 1: Express with a common denominator

To combine terms, identify a common denominator. The terms are \(a^{-1}\) and \(a^{-1}b^{-3}\). Notice that both terms can be expressed with the common denominator \(a^{-1}b^{-3}\). We express \(a^{-1}\) with this denominator as \(\frac{a^{-1}b^3}{b^3}\), and the second term remains \(\frac{a^{-1}}{a^{-1}b^{-3}}\).

Step 2: Combine the fractions

Add the two fractions you just expressed: \(\frac{a^{-1}b^3}{b^3} + \frac{a^{-1}}{a^{-1}b^{-3}}\). Since they have the same denominator, add the numerators: \(a^{-1}b^3 + a^{-1}\). So the expression becomes \(\frac{a^{-1}b^3 + a^{-1}}{b^{-3}}\).

Step 3: Substitute and simplify the numerators

Factor \(a^{-1}\) out of the numerator. \(a^{-1}(b^3 + 1)\) is the result. Substitute \(a^{-1}\) as \(\frac{1}{a}\) to simplify: \(\frac{1}{a}(b^3 + 1)\).

Step 4: Simplify the expression

Replace \(b^{-3}\) in the denominator as \(\frac{1}{b^3}\), changing the expression to \(\frac{b^3}{a(b^3 + 1)}\). This way, the exponents are positive and expressed as a single fraction.

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of the same number by itself. For example, when you see something like \(a^n\), it means that the base \(a\) is multiplied by itself \(n\) times. Exponents can be positive, negative, or even zero.
When you encounter negative exponents, it's helpful to remember this rule: a negative exponent means you take the reciprocal of the base. For instance, \(a^{-1}\) is equivalent to \(\frac{1}{a}\). This rule is crucial when you are tasked with simplifying expressions involving negative exponents.
In this exercise, the expression \(a^{-1} + a^{-1}b^{-3}\) contains negative exponents. Converting these to their positive forms is part of simplifying the expression to meet the problem's requirement of positive exponents only.

Common Denominator

When working with fractions, combining them often requires a common denominator. The common denominator allows you to directly add or subtract the fractions by making sure the "bottoms" (denominators) are the same.
In the provided exercise, the terms \(a^{-1}\) and \(a^{-1}b^{-3}\) need a common denominator to be added together. By expressing both terms as fractions with the least common denominator \(a^{-1}b^{-3}\), they can be effectively combined into a single fraction. This step is essential for simplifying equations that involve fractional expressions.

Simplifying Expressions

Simplifying expressions means making them easier to understand or work with. This often involves combining like terms, factoring, or reducing fractions to their simplest form.
In this exercise, simplication includes converting terms with negative exponents into positive exponents, and combining terms under a common denominator. The initial expression, \(a^{-1} + a^{-1}b^{-3}\), is simplified to \(\frac{1}{a}(b^3 + 1)\) by rewriting it with positive exponents and simplifying the fraction. Simplification not only makes expressions easier to read but also to work with in further calculations.

Factoring

Factoring is about breaking down expressions into smaller, more manageable parts, usually into products of numbers or expressions that multiply together to give the original expression.
In the context of this exercise, factoring focuses on extracting common terms from the numerator. Once the common denominator is established, the expression \(a^{-1}b^3 + a^{-1}\) is factored by taking \(a^{-1}\) as a common factor from both parts of the sum, resulting in \(a^{-1}(b^3 + 1)\).
This factored form is crucial as it directly leads to the simplified expression \(\frac{b^3}{a(b^3 + 1)}\) with positive exponents, satisfying the problem's requirement. Understanding how to factor can greatly assist in simplifying complex expressions and solving mathematical problems efficiently.