Problem 52 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{y^{-2}-x^{-2}}{y^{-2}+x^{-2}}$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Negative Exponents

Recall that a negative exponent means that the base is on the wrong side of the fraction line. Therefore, rewrite the expression by converting negative exponents to fractions: \( y^{-2} = \frac{1}{y^2} \) and \( x^{-2} = \frac{1}{x^2} \). Using this, the expression becomes: \( \frac{\frac{1}{y^2} - \frac{1}{x^2}}{\frac{1}{y^2} + \frac{1}{x^2}} \).

Step 2: Find a Common Denominator

In both the numerator and the denominator, identify the common denominator for the terms involved. The common denominator for \( \frac{1}{y^2} \) and \( \frac{1}{x^2} \) is \( x^2y^2 \).

Step 3: Rewrite the Numerator

Rewrite the numerator using the common denominator: \( \frac{1}{y^2} - \frac{1}{x^2} = \frac{x^2}{x^2y^2} - \frac{y^2}{x^2y^2} = \frac{x^2 - y^2}{x^2y^2} \).

Step 4: Rewrite the Denominator

Rewrite the denominator using the common denominator: \( \frac{1}{y^2} + \frac{1}{x^2} = \frac{x^2}{x^2y^2} + \frac{y^2}{x^2y^2} = \frac{x^2 + y^2}{x^2y^2} \).

Step 5: Cancel the Common Denominator

Since both the numerator and the denominator now have the same common denominator, they cancel each other out. This leaves us with: \( \frac{x^2 - y^2}{x^2 + y^2} \).

Key Concepts

Negative Exponents

When working with algebraic fractions, you might encounter negative exponents. A negative exponent, such as in the expression \( y^{-2} \), suggests that the base is on the wrong side of the fraction line. Converting it involves flipping the fraction. For instance, \( y^{-2} \) becomes \( \frac{1}{y^2} \), and \( x^{-2} \) becomes \( \frac{1}{x^2} \). This conversion simplifies handling exponents and expression manipulation. By practicing these transformations, you will gain ease in manipulating algebraic expressions and simplify complex problems.

Common Denominator

In order to perform operations like addition or subtraction in algebraic fractions, finding a common denominator is crucial. A common denominator is essentially a shared multiple of the denominators involved in a fraction. For example, if you have expressions \( \frac{1}{y^2} \) and \( \frac{1}{x^2} \) within a fraction, their common denominator is \( x^2y^2 \).
Without a common denominator, adding or subtracting terms can get complicated and difficult. With a common denominator, it becomes easier to combine terms, whether they're in the numerator or the denominator. Always keep an eye out for opportunities to find and use common denominators in algebraic fractions.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form, making them easier to work with. Once you acquire a common denominator, you can simply rewrite the fraction using it. Take \( \frac{x^2}{x^2y^2} - \frac{y^2}{x^2y^2} \) for example. Here, the common denominator allows for direct subtraction.
After simplification, terms on a common denominator often cancel out, as seen when both the numerator and denominator of our expression share \( x^2y^2 \). At this step, it becomes crucial to cancel out these terms to arrive at the simplest form, like \( \frac{x^2 - y^2}{x^2 + y^2} \). This not only cuts down on mathematics work but also reduces errors.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and arithmetic operations. In learning algebra, it's essential to understand how they form and change. These expressions can range from straightforward like \( x + 2 \) to more complex ones with fractions and exponents.
Mastery in simplifying these expressions with fractions and exponents is key to solving higher-level problems in algebra. Techniques like dealing with negative exponents, finding common denominators, and fraction simplification help in this process, transforming complex expressions into simpler, more manageable ones.
Keep practicing these skills to navigate through and understand the world of algebraic expressions better.