Problem 69 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{4}{c d}}{c^{-1}+d^{-1}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{4}{c+d} \).

Step 1: Rewrite Complex Fraction

First, observe the structure of the complex fraction: \( \frac{\frac{4}{cd}}{c^{-1} + d^{-1}} \). This expression consists of a fraction on top (numerator) and a sum of terms below (denominator).

Step 2: Simplify Denominator

Recognize that \( c^{-1} \) is equivalent to \( \frac{1}{c} \) and \( d^{-1} \) is equivalent to \( \frac{1}{d} \). Thus, the denominator becomes \( \frac{1}{c} + \frac{1}{d} \).

Step 3: Combine Denominator

Combine the terms in the denominator by finding a common denominator. The common denominator of \( c \) and \( d \) is \( cd \). Rewrite: \( \frac{1}{c} = \frac{d}{cd} \) and \( \frac{1}{d} = \frac{c}{cd} \). Therefore, \( \frac{1}{c} + \frac{1}{d} = \frac{d}{cd} + \frac{c}{cd} = \frac{c + d}{cd} \).

Step 4: Divide by the Fraction

To divide \( \frac{4}{cd} \) by \( \frac{c+d}{cd} \), multiply by the reciprocal. Thus, it is: \( \frac{4}{cd} \times \frac{cd}{c+d} \).

Step 5: Simplify Multiplication

Notice that \( cd \) in the numerator and denominator cancel out: \( \frac{4}{cd} \times \frac{cd}{c+d} = \frac{4}{c+d} \). So, the expression simplifies to \( \frac{4}{c+d} \).

Step 6: Conclusion

The complex fraction \( \frac{\frac{4}{cd}}{c^{-1} + d^{-1}} \) simplifies to \( \frac{4}{c+d} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is a vital skill in math that helps make calculations more manageable. A fraction consists of a numerator (the top part) and a denominator (the bottom part). In a complex fraction, both the numerator and the denominator can themselves be fractions.
Understanding how to handle these types of fractions involves steps that involve both simplifying the individual fractions and managing operations between them.

• First, identify any fractions within the numerator and the denominator. In the complex fraction \( \frac{\frac{4}{cd}}{c^{-1} + d^{-1}} \), the numerator itself is \( \frac{4}{cd} \).
• Next, simplify the denominators by recognizing algebraic equivalences, such as \( c^{-1} = \frac{1}{c} \).
• Combine terms using a common denominator, such as \( cd \) for \( c^{-1} + d^{-1} \), to simplify the denominator into a single fraction.
• Multiply by the reciprocal of the denominator’s simplified form, allowing the expression to simplify through multiplication.

These steps form a consistent pattern for tackling many types of complex fractions.

Algebraic Expressions

Algebraic expressions involve combinations of numbers, variables, and arithmetic operations. They are a core part of algebra and are essential for understanding and solving equations.
When simplifying complex fractions like \( \frac{\frac{4}{cd}}{c^{-1}+d^{-1}} \), it involves an understanding of how variables interact in expressions.

• Recognize that variables such as \( c \) or \( d \) can act as placeholders for any number, allowing for general solutions.
• Understand how exponents and negative exponents work. For example, \( c^{-1} \) is another way to express \( \frac{1}{c} \).
• Learn to add expressions with unlike denominators by finding and using a common denominator. Here, \( c \) and \( d \) combine as \( \frac{c+d}{cd} \).

Algebraic expressions simplify and provide pathways to solutions when properly applied to complex equations.

Reciprocals

Reciprocals are an important concept in mathematics, especially when simplifying fractions. The reciprocal of a number or expression is what you multiply it by to get the value 1.
In the context of complex fractions, knowing how to find and use reciprocals is crucial. Consider the problem \( \frac{\frac{4}{cd}}{c^{-1} + d^{-1}} \):

• First, identify the denominator as a single fraction: \( \frac{c+d}{cd} \).
• The reciprocal of this fraction is \( \frac{cd}{c+d} \), because multiplying \( \frac{c+d}{cd} \) by \( \frac{cd}{c+d} \) yields 1.
• Use this reciprocal to effectively "cancel out" the denominator, simplifying the whole expression. Multiplying \( \frac{4}{cd} \) by \( \frac{cd}{c+d} \) gives \( \frac{4}{c+d} \).

Thus, a clear understanding of how and when to use reciprocals simplifies division among fractions and is essential for solving complex fraction problems.