Problem 75 - Algebra and Trigonometry

Question

\(71-76\) m simplify the expression. (This type of expression arises in calculus when using the "quotient rule.") $$ \frac{3(1+x)^{1 / 3}-x(1+x)^{-2 / 3}}{(1+x)^{2 / 3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3 + 2x}{(1+x)^{2/3}} \).

Step 1: Understand the Problem

We have an expression with the form \( \frac{3(1+x)^{1/3}-x(1+x)^{-2/3}}{(1+x)^{2/3}} \). Our goal is to simplify this expression.

Step 2: Combine Terms with a Common Denominator

Notice that both terms in the numerator can be combined over a common denominator, \((1+x)^{2/3}\). This results in: \( \frac{3(1+x)^{1/3}(1+x)^{2/3} - x}{(1+x)^{2/3}} \).

Step 3: Simplify the Combined Expression

Simplify \((1+x)^{1/3}(1+x)^{2/3}\) to \((1+x)^{1}\), which gives us \( \frac{3(1+x) - x}{(1+x)^{2/3}} \).

Step 4: Simplify the Numerator

Distribute in the numerator: \(3(1+x) - x = 3 + 3x - x = 3 + 2x\). This simplifies the expression to \( \frac{3 + 2x}{(1+x)^{2/3}} \).

Step 5: Simplified Result

The simplified expression is \( \frac{3 + 2x}{(1+x)^{2/3}} \).

Key Concepts

Quotient Rule

In calculus, understanding how to handle different types of expressions is crucial, especially when they involve fractions of functions. The quotient rule is a powerful differentiation technique useful for functions expressed as the quotient of two differentiable functions. If you have a function \(u/v\) where both \(u\) and \(v\) are functions of \(x\), the quotient rule helps you find its derivative. The rule is given by:

• \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \)

This method simplifies the differentiation process, avoiding errors associated with attempting to break apart the quotient directly. Understanding this rule can deeply enhance your ability to tackle complex calculus problems without needing to manually simplify the entire expression first.

Simplifying Expressions

Simplifying mathematical expressions requires combining like terms and reducing the expression to its simplest form. This involves performing operations such as combining or canceling terms, factoring, and applying mathematical properties like the distributive property. For instance, in the given problem:

• We simplify \((1+x)^{1/3}(1+x)^{2/3}\) by recognizing the property that allows us to add the exponents when multiplying. This results in \((1+x)^1\).

Other simplicity checks may involve recognizing common patterns in an equation or term distribution. Beginning with a clear understanding of the goal helps underscore the importance of arriving at the most straightforward form of the expression, making it more manageable to work with in calculus.

Numerator and Denominator

In any fraction, the number above the line is the numerator, and the number below is the denominator. Correct manipulation of these two components is essential in both calculus and basic arithmetic. For the provided expression, we dealt with both separately before bringing them back together.
The steps taken are:

• Combining terms in the numerator over a shared denominator.
• Distributing and simplifying the numerator itself \((3(1+x) - x) = 3 + 2x\).
• Reintegrating that simplified numerator over the original denominator \((1+x)^{2/3}\).

By handling each part carefully, complex problems become simpler, ensuring accurate calculations and progression in solving mathematical tasks.

Fractional Exponents

Fractional exponents can initially appear cumbersome, yet they provide a powerful way to represent roots and powers combined in a single expression. Consider a fractional exponent such as \((1+x)^{1/3}\). This represents the cube root of \((1+x)\). When dealing with expressions involving fractional exponents, knowing how to manipulate them is essential.
Here are some basic rules:

• \((a^m)^{1/n} = a^{m/n}\), which simplifies expressions to their simplest form.
• When multiplying terms with the same base, add the exponents: \((a^{1/3})(a^{2/3}) = a^{(1/3)+(2/3)} = a^1\).

In the given calculus problem, these key rules helped us simplify the expression efficiently. Understanding fractional exponents enables you to break down complex expressions while maintaining mathematical accuracy.