Problem 73 - Algebra and Trigonometry

Question

Simplify the compound fractional expression. $$ \frac{x^{-2}+y^{-2}}{x^{-1}+y^{-1}} $$

Step-by-Step Solution

Answer

\( \frac{x^2 + y^2}{xy(x + y)} \)

Step 1: Understand the Expression

The expression is a compound fractional expression consisting of an overall fraction with both the numerator and the denominator being sums of terms with negative exponents.

Step 2: Simplify the Numerator

The numerator is given as \( x^{-2} + y^{-2} \). Terms with negative exponents can be rewritten as positive exponents in the denominator. Therefore, the numerator can be rewritten as \( \frac{1}{x^2} + \frac{1}{y^2} \). To simplify, find a common denominator, which is \( x^2y^2 \).\[ \frac{1}{x^2} + \frac{1}{y^2} = \frac{y^2}{x^2y^2} + \frac{x^2}{x^2y^2} = \frac{y^2 + x^2}{x^2y^2} \].

Step 3: Simplify the Denominator

Similarly, simplify the denominator \( x^{-1} + y^{-1} \). This can be rewritten using the reciprocal as \( \frac{1}{x} + \frac{1}{y} \). To simplify, find a common denominator, which is \( xy \).\[ \frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy} \].

Step 4: Combine and Simplify the Compound Fraction

The expression now is \[ \frac{\frac{y^2 + x^2}{x^2y^2}}{\frac{y + x}{xy}} \]. To simplify, multiply by the reciprocal of the denominator:\[ \frac{y^2 + x^2}{x^2y^2} \times \frac{xy}{y + x} \].

Step 5: Simplify the Resulting Fraction

Multiply the numerators and the denominators: \[ \frac{(y^2 + x^2)xy}{x^2y^2(y + x)} \] The \( xy \) terms in the numerator and denominator cancel out:\[ \frac{y^2 + x^2}{xy(y + x)} \].

Step 6: Final Expression

The simplified form of the compound fractional expression is: \[ \frac{x^2 + y^2}{xy(x + y)} \].

Key Concepts

Negative Exponents

Negative exponents can seem a little tricky at first, but they're actually quite straightforward once you understand the rule that defines them. A negative exponent means that the base will be on the opposite side of the fraction bar. For example, \( x^{-2} \) is equivalent to \( \frac{1}{x^2} \). This is because a negative exponent indicates that you have a reciprocal. The crucial thing to remember is:

• \( a^{-n} = \frac{1}{a^n} \)

Using this rule, you can simplify expressions and get rid of negative exponents by converting them to a positive exponent in the denominator. It helps in making the expressions more manageable and easier to understand, as you deal with positive integers that we're more familiar with.
Whenever you encounter negative exponents, just remember to flip it. This will make simplifying complex or compound expressions much smoother.

Compound Fractions

Compound fractions, often called complex fractions, involve fractions within fractions, either in the numerator, denominator, or both. They can look puzzling at first, but with a step-by-step approach, they can be simplified easily.
To simplify a compound fraction:

• First, simplify the fractions in the numerator and denominator as much as possible.
• Find a common denominator for any fractions you identify within the fraction.
• Rewrite the fractions as a single fraction for the numerator and another for the denominator.
• Convert the compound fraction by multiplying by the reciprocal of the denominator fraction.

Start by transforming each part into a simple fraction and then use basic fraction rules for further simplification. This methodical approach reduces compound fractions to their simplest forms, making them easier to handle.

Reciprocals

Reciprocals are fundamental in fraction manipulation and simplification, including when working with negative exponents and compound fractions. A reciprocal of a number or expression is essentially flipping the numerator and the denominator.

• The reciprocal of \( a \) is \( \frac{1}{a} \).
• If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).

Reciprocals come into play frequently when you need to divide by a fraction or to simplify a complex fraction. To divide by a fraction, you multiply by its reciprocal because multiplying by the reciprocal of a fraction is equivalent to dividing by that fraction.
In the simplification of compound fractions, we multiply by the reciprocal of the fraction found in the denominator. This technique is a powerful tool for simplifying algebraic expressions and solving equations more effortlessly.