Problem 69 - Intermediate Algebra

Question

Simplify each expression. \(\frac{x^{-2}+y^{-2}}{x^{-2}-y^{-2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{x^2 + y^2}{x^2 - y^2}\).

Step 1: Recognize Negative Exponents

Remember that negative exponents indicate the reciprocal of the base raised to the positive exponent. Thus, \(x^{-2} = \frac{1}{x^2}\) and \(y^{-2} = \frac{1}{y^2}\). Substitute these into the expression.

Step 2: Rewrite the Expression

Substitute the negative exponents in the expression: \(\frac{x^{-2} + y^{-2}}{x^{-2} - y^{-2}}\) becomes \(\frac{\frac{1}{x^2} + \frac{1}{y^2}}{\frac{1}{x^2} - \frac{1}{y^2}}\).

Step 3: Find a Common Denominator for the Numerator

In the numerator, find a common denominator, which is \(x^2y^2\), resulting in \(\frac{y^2}{x^2y^2} + \frac{x^2}{x^2y^2}\), which simplifies to \(\frac{y^2 + x^2}{x^2y^2}\).

Step 4: Find a Common Denominator for the Denominator

Similarly, in the denominator, find a common denominator \(x^2y^2\), so \(\frac{y^2}{x^2y^2} - \frac{x^2}{x^2y^2}\) simplifies to \(\frac{y^2 - x^2}{x^2y^2}\).

Step 5: Divide Fractions

Divide the fractions from Step 3 and Step 4: \(\frac{\frac{y^2 + x^2}{x^2y^2}}{\frac{y^2 - x^2}{x^2y^2}}\). In division of fractions, multiply by the reciprocal of the denominator: \(\frac{y^2 + x^2}{x^2y^2} \times \frac{x^2y^2}{y^2 - x^2}\).

Step 6: Simplify the Expression

The \(x^2y^2\) terms cancel out, leading to: \(\frac{y^2 + x^2}{y^2 - x^2}\).

Step 7: Rearrange Terms

Rearrange the terms to match conventional forms: \(\frac{x^2 + y^2}{-(x^2 - y^2)}\), which simplifies to \(-\frac{x^2 + y^2}{x^2 - y^2}\).

Key Concepts

Negative Exponents

A negative exponent tells you to take the reciprocal of the base and then raise it to the opposite positive power. This is a fundamental concept in simplification. Understanding this can help you rewrite complex expressions such as those with negative exponents. Consider the negative exponent in the expression you worked on. When you see

• \(x^{-2}\)

this is equivalent to taking the inverse of \(x\) squared, thus:\( \frac{1}{x^2} \). Similarly,

• \(y^{-2} = \frac{1}{y^2}\).

This way of rearranging helps simplify further operations, particularly when dealing with fractions.
Trying to visualize these exponents as fractions helps create a clear path for simplification. By replacing the negative exponents with these fractions, you move forward in making the entire expression less cumbersome.

Common Denominator

Finding a common denominator is a crucial step when simplifying expressions involving fractions. This doesn't just apply to rational expressions, but to many types of problems dealing with addition and subtraction of fractions. For example, if you take

• \(\frac{1}{x^2} + \frac{1}{y^2}\)

you would need denominators to be the same to perform the addition. Here, the common denominator would be

• \(x^2 y^2\)

This requires you to rewrite each fraction:\(\frac{y^2}{x^2 y^2} + \frac{x^2}{x^2 y^2}\)So you end up with one fraction rather than two:\(\frac{x^2 + y^2}{x^2 y^2}\)The process remains the same in the denominator. Ensuring that both fractions you are working with have common denominators simplifies complex operations like division later.

Fraction Division

Dividing fractions is an essential skill that comes into play often in algebraic expressions. Understanding that dividing by a fraction is the same as multiplying by its reciprocal can make this process intuitive. Take the final step in the solution: dividing

• \(\frac{\frac{x^2 + y^2}{x^2 y^2}}{\frac{y^2 - x^2}{x^2 y^2}}\)

When you divide by a fraction, you multiply by the reciprocal instead. Hence,

• \(\frac{x^2 + y^2}{x^2 y^2} \times \frac{x^2 y^2}{y^2 - x^2}\)

This operation immediately simplifies your process, allowing you to cancel the common terms in the numerators and denominators:
\(x^2y^2\) cancels itself out, leaving you with

• \(\frac{x^2 + y^2}{y^2 - x^2}\).

Knowing to multiply by the reciprocal simplifies the task and works on a similar logic as multiplication.