Problem 58 - Algebra for College Students

Question

Simplify. $$\frac{a+27 a^{-2}}{1-3 a^{-1}+9 a^{-2}}$$

Step-by-Step Solution

Answer

a + 3.

Step 1: Simplify the numerator

Rewrite the numerator \(a + 27a^{-2}\) by factoring out the common term \(a^{-2}\). \[\frac{a + 27a^{-2}}{1 - 3a^{-1} + 9a^{-2}} = \frac{a^{-2} (a^3 + 27)}{1 - 3a^{-1} + 9a^{-2}}.\]

Step 2: Simplify the denominator

Rewrite the denominator \(1 - 3a^{-1} + 9a^{-2}\) as a quadratic in terms of \(a^{-1}\), i.e., \(a^{-2} (a^2 - 3a + 9).\)

Step 3: Factor if possible

Recognize that \(a^3 + 27 = (a + 3)(a^2 - 3a + 9)\) using the sum of cubes formula.

Step 4: Cancel common terms

The expression simplifies to \[\frac{a^{-2} (a + 3)(a^2 - 3a + 9)}{a^{-2} (a^2 - 3a + 9)}.\] Cancel the common terms \(a^{-2} (a^2 - 3a + 9)\), leaving \[a + 3.\]

Step 5: Final simplified form

The final simplified expression is \(a + 3\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex expressions into simpler forms. This makes them easier to understand and work with.
One common method is to look for common factors that can be factored out. In the given exercise, we start by simplifying the numerator and denominator separately.
We can factor out common terms to make the expressions easier to handle.

• Numerator: The expression is rewritten as two separate terms in terms of their exponents.
• Denominator: The expression is treated as a quadratic equation in the variable of the negative exponent.

This step-by-step approach helps in reducing errors and understanding the structure of the expression.

Factoring

Factoring is the process of breaking down an algebraic expression into simpler parts called factors that, when multiplied together, give the original expression. It is essential for simplifying expressions.
In the given problem, we factor out the common term \(a^{-2}\) from both the numerator and denominator.
This step is crucial as it allows us to recognize and cancel out common factors later.

• Identify common factors in both the numerator and denominator.
• Use algebraic identities, such as the sum of cubes formula, to simplify the expression further.

Factoring can significantly simplify complex expressions and is a key skill in algebra.

Sum of Cubes

The sum of cubes formula is an algebraic identity that states: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
This formula helps to simplify expressions involving the sum of two cubed terms.

• In the exercise, \(a^3 + 27\) is identified as a sum of cubes since \(27 = 3^3\).
• Applying the formula, we rewrite it as \( (a + 3)(a^2 - 3a + 9)\).

Recognizing and using this formula makes it possible to factor complex expressions into simpler polynomials. This simplification is essential for further algebraic manipulations and solving equations.