Problem 51 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{y^{-1}+x^{-1}}{(x y)^{-1}}$$

Step-by-Step Solution

Answer

The simplified expression is \(x + y\).

Step 1: Simplify the Numerator

The expression in the numerator is given by \( y^{-1} + x^{-1} \). Let's rewrite these terms as fractions: \( y^{-1} = \frac{1}{y} \) and \( x^{-1} = \frac{1}{x} \). So, the numerator becomes \( \frac{1}{y} + \frac{1}{x} \). We need a common denominator to add these fractions, which is \( xy \). This gives us:\[ \frac{1}{y} + \frac{1}{x} = \frac{x}{xy} + \frac{y}{xy} = \frac{x+y}{xy}. \]

Step 2: Simplify the Denominator

The expression in the denominator is \((xy)^{-1}\). Rewrite this as a fraction: \((xy)^{-1} = \frac{1}{xy}\).

Step 3: Divide the Numerator by the Denominator

Now, divide the simplified numerator by the simplified denominator:\[ \frac{\frac{x+y}{xy}}{\frac{1}{xy}}. \]When dividing by a fraction, you multiply by its reciprocal:\[ = \frac{x+y}{xy} \times xy. \]

Step 4: Simplify the Expression

The \(xy\) terms in the numerator and denominator cancel out when multiplied:\[ \frac{x+y}{xy} \cdot xy = x+y. \]Thus, the simplified expression is \(x + y\).

Key Concepts

Negative Exponents

Negative exponents might seem a little intimidating at first, but they are quite straightforward once you understand the concept. Essentially, a negative exponent indicates that you should take the reciprocal of the base and raise it to the corresponding positive exponent. For example, when you see an expression like \( y^{-1} \), this is equivalent to \( \frac{1}{y} \). The idea with negative exponents:

• Take the reciprocal of the base.
• Change the exponent from negative to positive.

By applying this rule, any negative exponent in an expression can be rewritten as a fraction. This is a vital step when simplifying rational expressions with negative exponents.

Common Denominator

Let's talk about common denominators. You might remember this from adding fractions in basic arithmetic. A common denominator is used to join fractions, making it easier to perform addition or subtraction.
When dealing with fractions like \( \frac{1}{y} \) and \( \frac{1}{x} \), you need a common denominator to add them together effectively.
In our expression, the common denominator here is \( xy \). This is because each fraction needs to have the same base in order to be combined:

• Multiply the numerator of each fraction with the denominator of the other fraction.
• Adapt both fractions to this common denominator, \( xy \).

This will transform \( \frac{1}{y} + \frac{1}{x} \) into a single fraction, \( \frac{x+y}{xy} \), paving the way for further simplification.

Fraction Multiplication

Fraction multiplication is straightforward once you get the hang of it. When you multiply a fraction, you simply multiply the numerators together and the denominators together. With something like dividing fractions, you are actually doing a sort of multiplication, but with a twist.
Consider the expression \( \frac{\frac{x+y}{xy}}{\frac{1}{xy}} \). When dividing by a fraction, you actually multiply by its reciprocal. For instance:

• First, identify the reciprocal of the fractional divisor.
• In this case, multiplying \( \frac{x+y}{xy} \) by \( xy \) (the reciprocal of \( \frac{1}{xy} \)).
• This neutralizes the denominator, simplifying the expression to \( x+y \).

Understanding fraction multiplication in this context enables seamless simplification of complex rational expressions.

Algebraic Simplification

Algebraic simplification helps you streamline expressions to make them more manageable. It involves performing operations that reduce an expression to its simplest form. In our context, this often involves canceling common factors or using identities to your advantage.
With our expression, \( \frac{x+y}{xy} \times xy \), the \( xy \) in the numerator and the denominator cancel each other out naturally:

• This eliminates the rational components, reducing it to a simple \( x+y \).

Simplifying rational expressions often boils down to strategically combining these techniques, allowing complex fractions to be expressed in simpler terms.