Problem 61 - Algebra and Trigonometry

Question

\(55-64=\) Simplify the compound fractional expression. $$ \frac{x^{-2}-y^{-2}}{x^{-1}+y^{-1}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{y-x}{xy} \).

Step 1: Simplify the Numerator

The numerator of the expression is \( x^{-2} - y^{-2} \). This can be rewritten using the property of negative exponents: \( x^{-2} = \frac{1}{x^2} \) and \( y^{-2} = \frac{1}{y^2} \). Thus, the numerator becomes: \( \frac{1}{x^2} - \frac{1}{y^2} \). To combine these fractions, find a common denominator, which is \( x^2y^2 \), and rewrite the expression as \( \frac{y^2 - x^2}{x^2 y^2} \).

Step 2: Simplify the Denominator

The denominator is \( x^{-1} + y^{-1} \). Again, rewrite using negative exponents: \( x^{-1} = \frac{1}{x} \) and \( y^{-1} = \frac{1}{y} \). The expression becomes \( \frac{1}{x} + \frac{1}{y} \). Find a common denominator \( xy \), and rewrite as \( \frac{y + x}{xy} \).

Step 3: Combine Simplified Expressions

Now we have \( \frac{y^2 - x^2}{x^2 y^2} \) over \( \frac{y + x}{xy} \). Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, the original expression becomes: \[ \frac{y^2 - x^2}{x^2 y^2} \times \frac{xy}{y + x} \].

Step 4: Factor and Simplify

Factor \( y^2 - x^2 \) as \( (y-x)(y+x) \). The expression now is: \[ \frac{(y-x)(y+x) \cdot xy}{x^2 y^2 (y+x)} \]. The \( y+x \) terms cancel out, leaving: \[ \frac{(y-x) \cdot x}{x^2 y} \]. Simplify this expression to \( \frac{y-x}{x y} \).

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they follow a straightforward rule: they indicate the reciprocal of the base raised to the positive of the exponent. For example, if you have a term like \( x^{-2} \), it is equivalent to \( \frac{1}{x^2} \). This means that instead of multiplying the base by itself, you divide 1 by the base multiplied by itself.
\( x^{-2} \) becomes \( \frac{1}{x^2} \), and \( y^{-2} \) becomes \( \frac{1}{y^2} \).
Understanding this rule helps in simplifying expressions like \( x^{-1} + y^{-1} \) to \( \frac{1}{x} + \frac{1}{y} \). When you encounter negative exponents, always think of flipping the base and exponent to rewrite them as fractions.

Common Denominator

Combining fractions requires identifying a common denominator. This is the same principle you use when adding or subtracting fractions. A common denominator is a shared multiple of the denominators of your fractions.
In our example, to subtract \( \frac{1}{x^2} \) and \( \frac{1}{y^2} \), you need a least common denominator \( x^2y^2 \). This lets you combine the fractions: \( \frac{y^2}{x^2y^2} - \frac{x^2}{x^2y^2} = \frac{y^2 - x^2}{x^2y^2} \).
Always look at the denominators of your fractions. Once you've found the common denominator, rewrite each fraction with this denominator and then proceed with your addition or subtraction.

Factoring Polynomials

Factoring polynomials is a key skill in simplifying expressions, and it often involves recognizing special patterns. One such pattern is the difference of squares, displayed as \( a^2 - b^2 = (a-b)(a+b) \).
This was used in the solution where \( y^2 - x^2 \) was factored as \( (y-x)(y+x) \). Recognizing these patterns allows you to break polynomials into their simpler components, which makes further simplification easier.
Factoring can also involve identifying and extracting common factors, but always look for these special patterns first. They help you condense and simplify polynomial expressions efficiently.