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Magazine Chapter 1: Problem 51
Simplify the expression. $$\frac{y^{-1}+x^{-1}}{(x y)^{-1}}$$
Short Answer
Expert verified
The simplified expression is \(x + y\).
Step by step solution
01
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:\[ \frac{1}{y} + \frac{1}{x} = \frac{x}{xy} + \frac{y}{xy} = \frac{x+y}{xy}. \]
02
Simplify the Denominator
The expression in the denominator is \((xy)^{-1}\). Rewrite this as a fraction: \((xy)^{-1} = \frac{1}{xy}\).
03
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:\[ = \frac{x+y}{xy} \times xy. \]
04
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 \(x + y\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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:
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, \( xy \).
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:
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 \( x+y \).
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:
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 \( x+y \).
