Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operations and simplify.\(\frac{\left[\frac{x^{2}}{(x+1)^{2}}\right]}{\left[\frac{x}{(x+1)^{3}}\right]}\)

Step-by-Step Solution

Answer

\((x+1)^{2}\) is the simplified form of the given expression.

Step 1: Write the given expression

The given complex fraction is \(\frac{\left[\frac{x^{2}}{(x+1)^{2}}\right]}{\left[\frac{x}{(x+1)^{3}}\right]}\)

Step 2: Division of fractions

A division of fractions can be turned into a multiplication if we flip (take the reciprocal of) the fraction that is being divided. Therefore, we will multiply the first fraction by the reciprocal of the second fraction: \(\frac{x^{2}}{(x+1)^{2}}*\frac{(x+1)^{3}}{x}\)

Step 3: Simplification

Let's simplify the expression. The \(x\) in the numerator and denominator cancels out, and only one \(x+1\) in the denominator cancels out, leaving two (i.e., \( (x+1)^2 \) behind. So, the simplified expression becomes \((x+1)^{2}\)

Key Concepts

Division of Fractions

When we deal with complex fractions, the key operation often required is the division of fractions. Imagine a fraction where both the numerator and the denominator are themselves fractions. It's like a division problem that involves two smaller fractions.
To handle complex fractions, we'll use the principle that dividing by a fraction is the same as multiplying by its reciprocal. This means we flip the second fraction (the denominator) upside down and change the division sign to multiplication.

• If we start with a complex fraction like \( \frac{\left[\frac{a}{b}\right]}{\left[\frac{c}{d}\right]} \), we can rewrite it as \( \frac{a}{b} * \frac{d}{c} \).
• This process changes the problem from division, which can be daunting, to multiplication, which is often simpler to perform and understand.

This transformation helps to clarify the operations and make the solution path more straightforward.

Simplification

After setting up our fractions for multiplication, the next critical step is simplification. Simplification helps to reduce the expression to its most basic form, making it easier to understand and work with. Here are some steps to guide you:

• First, look for any common factors in the numerator and the denominator that can be divided out.

Let's take an example to illustrate this. In the expression \( \frac{x^{2}}{(x+1)^{2}} \times \frac{(x+1)^{3}}{x} \), we simplify by:

• Canceling one \(x\) from the \(x^{2}\) in the numerator and \(x\) in the denominator.
• Canceling \((x+1)^2\) from \((x+1)^3\) in the numerator and \((x+1)^2\) in the denominator.

These reduce our expression to \( (x+1)^{2} \). By removing these common factors, we ensure the expression is clean and easy to interpret.

Reciprocal

A reciprocal is a special type of inverse operation. In the context of division in fractions, the reciprocal is what turns a division problem into multiplication.

• The reciprocal of a number is simply one over that number. For example, the reciprocal of \( a \) is \( \frac{1}{a} \).

When dealing with fractions, finding the reciprocal involves "flipping" the numerator and the denominator. So, if you have \( \frac{c}{d} \), its reciprocal would be \( \frac{d}{c} \).
This flipping is crucial because it transforms division into multiplication, which can then be handled using more familiar rules and techniques. Understanding the role of reciprocals in dividing fractions can ease the complexity and ensure you approach such problems with confidence.