Problem 37 - Prealgebra and Introductory Algebra

Question

True or false? If the denominator of a complex fraction is the reciprocal of the numerator, then the complex fraction is equal to the square of its numerator.

Step-by-Step Solution

Answer

The statement is true, if the denominator of a complex fraction is the reciprocal of the numerator, then the complex fraction is indeed equal to the square of its numerator.

Step 1: Understand the problem

Let's denote the numerator of the complex fraction as x. Thus, the fraction will be \(\frac{x}{1/x}\). Our task is to verify if \(\frac{x}{1/x}\) equals \(x^2\).

Step 2: Simplify the fraction

To simplify a fraction where the denominator is the reciprocal of the numerator, we multiply the numerator and denominator. Hence, this will become \(x * x\) which equals \(x^2\).

Step 3: Compare to the square of the numerator

On simplifying, we got \(x^2\) which is the square of the numerator, x. This proves that the complex fraction is indeed equal to the square of its numerator when the denominator is the reciprocal of the numerator.

Key Concepts

Reciprocal

Understanding the concept of reciprocal is crucial when dealing with complex fractions. The reciprocal of a number is essentially what you get when you divide 1 by that number. For example, the reciprocal of a number \(x\) is \(\frac{1}{x}\). This concept also applies to fractions and algebraic expressions.

• The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).
• Reciprocals are vital for division and simplification tasks, especially in algebra.

When you multiply a number by its reciprocal, the result is always 1. This property is instrumental in simplifying expressions like complex fractions.

Numerator

The numerator is the top part of a fraction. It represents how many parts of a whole are being considered. In the case of the fraction \(\frac{x}{1/x}\) from the exercise, the numerator is \(x\). This is crucial because any transformation or operation on a fraction directly affects its numerator.

• In algebra, the numerator can be either a constant or a variable expression.
• Understanding the structure of a numerator helps in effectively managing operations involving fractions.

For complex fractions, recognizing the numerator's role will guide you in appropriate algebraic manipulation.

Denominator

The denominator is the bottom part of a fraction. It signifies the total number of equal parts that make up a whole, acting as a divisor for the numerator. In our exercise, the denominator is the reciprocal of the numerator, \(\frac{1}{x}\). This forms a complex fraction where division by the reciprocal leads to simpler calculations.

• A proper grasp of the denominator's role aids in manipulating and simplifying fractions.
• In algebra, denominators can also be variables or expressions, adding layers to simplification.

In complex fractions, the denominator plays a pivotal role in determining how simplification unfolds.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest forms. With complex fractions, simplification often involves manipulating the numerator and the denominator by using properties of reciprocals.

• Using reciprocals allows us to multiply instead of dividing, which often simplifies the calculations.
• When both the numerator and the denominator are algebraic expressions, understanding their interaction is the key to successful simplification.

In our exercise, simplifying \(\frac{x}{1/x}\) by multiplying \(x\) and \(x\) results in \(x^2\), demonstrating simplification in action. This process not only clarifies the expression but also proves the original statement in the exercise.