Problem 49 - Algebra for College Students

Question

Simplify. $$\frac{x^{-1}+x^{-2}}{x+x^{-2}}$$

Step-by-Step Solution

Answer

\(\frac{x+1}{x^3+1}\)

Step 1: Simplify the numerator

The numerator is given as \ \ \(x^{-1} + x^{-2}\). Rewrite each term using positive exponents: \ \ \(\frac{1}{x} + \frac{1}{x^2}\).

Step 2: Simplify the denominator

The denominator is given as \ \ \(x + x^{-2}\). Rewrite each term using positive exponents: \ \ \(x + \frac{1}{x^2}\).

Step 3: Find a common denominator for the numerator

The common denominator for \ \ \(\frac{1}{x} + \frac{1}{x^2}\) is \ \ \(x^2\). Rewrite each term: \ \ \(\frac{x}{x^2} + \frac{1}{x^2}\), which simplifies to \ \ \(\frac{x+1}{x^2}\).

Step 4: Find a common denominator for the denominator

The common denominator for \ \ \(x + \frac{1}{x^2}\) is \ \ \(x^2\). Rewrite each term: \ \ \(\frac{x^3}{x^2} + \frac{1}{x^2}\), which simplifies to \ \ \(\frac{x^3+1}{x^2}\).

Step 5: Simplify the entire expression

Divide the simplified numerator by the simplified denominator: \ \ \(\frac{\frac{x+1}{x^2}}{\frac{x^3+1}{x^2}} = \frac{x+1}{x^3+1}\).

Key Concepts

Understanding Negative Exponents

Negative exponents can turn a complicated-looking expression into something more manageable. When we see an expression like \(x^{-n}\), it means the reciprocal of \(x\) raised to the positive \(n\). For example, \(x^{-1}\) becomes \(\frac{1}{x}\) and \(x^{-2}\) becomes \(\frac{1}{x^2}\). Whenever you encounter a negative exponent, just remember it's another way of writing fractions. This method will help you simplify many algebraic expressions by making exponents positive.

Common Denominators

A common denominator is essential for adding or subtracting fractions. It makes the bottom part (denominator) of the fractions the same, so you can easily combine them. For instance, between \(\frac{1}{x}\) and \(\frac{1}{x^2}\), the common denominator is \(x^2\). To rewrite each fraction, we get \(\frac{x}{x^2} + \frac{1}{x^2}\), which then simplifies to \frac{x+1}{x^2}\. Always remember that finding the least common denominator makes the process smoother.

Fraction Division

Dividing fractions might look tricky, but it's straightforward once you know the trick. The rule for dividing fractions is to multiply by the reciprocal of the divisor. Let's take our simplified problem as an example: \(\frac{\frac{x+1}{x^2}}{\frac{x^3+1}{x^2}}\). Instead of dividing by \(\frac{x^3+1}{x^2}\), you multiply by its reciprocal: \(\frac{x^2}{x^3+1}\). This gives you \frac{x+1}{x^3+1}\. By simplifying expressions through division, you can reduce them to their simplest form.