Problem 76 - Algebra and Trigonometry

Question

\(71-76\) m simplify the expression. (This type of expression arises in calculus when using the "quotient rule.") $$ \frac{(7-3 x)^{1 / 2}+\frac{3}{2} x(7-3 x)^{-1 / 2}}{7-3 x} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{(7-3x)^{1/2}}\).

Step 1: Identify Components

The given expression is \( \frac{(7-3x)^{1/2} + \frac{3}{2}x(7-3x)^{-1/2}}{7-3x} \). Break down this expression into its components. The numerator is \((7-3x)^{1/2} + \frac{3}{2}x(7-3x)^{-1/2}\) and the denominator is \(7-3x\).

Step 2: Simplify the Numerator

Focus on simplifying the numerator: \((7-3x)^{1/2} + \frac{3}{2}x(7-3x)^{-1/2}\). Combine these terms over a common denominator: \(\frac{(7-3x)(7-3x)^{-1/2} + \frac{3}{2}x}{(7-3x)^{-1/2}}\). This simplifies to \(\frac{(7-3x)^{1/2} + \frac{3}{2}x}{(7-3x)^{-1/2}}\).

Step 3: Apply Common Denominator

Multiply numerator terms by \((7-3x)^{1/2}\) over itself to have a common denominator, turning the expression into \(\frac{\left((7-3x) + \frac{3}{2}x\right)}{(7-3x)^{1/2}}\). The numerator becomes \((7-3x) + \frac{3}{2}x\).

Step 4: Combine Like Terms in the Numerator

Combine the terms in the new numerator: \((7-3x) + \frac{3}{2}x = 7 - \frac{3}{2}x\).

Step 5: Divide by the Denominator

Divide the entire simplified numerator by the denominator, essentially moving \((7-3x)^{1/2}\) in the denominator of the overall expression. Since \((7-3x)\) also appears, cancel out terms where possible. This reduces the expression to \(\frac{1}{(7-3x)^{1/2}}\).

Step 6: Final Simplification

Reevaluate any potential simplifications or cancellations, ensuring all terms align and simplify as required, which renders a final simple form of the expression: \(\frac{1}{(7-3x)^{1/2}}\).

Key Concepts

Quotient Rule

When studying calculus, one vital concept is the quotient rule. This rule helps us differentiate equations that involve fractions, particularly where both the numerator and denominator are functions of a variable. The quotient rule formula is:

• If you have a function represented as \[ \frac{f(x)}{g(x)} \], then its derivative, using the quotient rule, is \[ \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \].

The formula derives from applying the product and chain rules of differentiation. In the problem above, understanding this rule is a precursor step, often guiding the simplification process in such calculus exercises.

Simplification

Simplification in mathematics is about making expressions easier to work with, without changing their value. In calculus, this is particularly important for ease of calculation and to avoid errors. In the provided exercise, simplification is achieved by:

• Breaking down complex expressions into more manageable pieces.
• Using algebraic techniques such as combining terms, applying common denominators, and factoring out common elements.

This process helps highlight underlying structures within an equation, allowing for further manipulations or solving. Simplification is a skill that often depends on practice and a good understanding of algebraic principles.

Numerator and Denominator

The terms numerator and denominator are fundamental, especially when dealing with fractional expressions in calculus. The numerator is the top part of a fraction, while the denominator is the bottom part. In the given exercise:

• The numerator was \[ (7 - 3x)^{1/2} + \frac{3}{2}x(7-3x)^{-1/2} \], consisting of several terms needing simplification.
• The denominator is simply \[ 7 - 3x \], which must be considered when simplifying the expression.

Understanding these components is crucial as it aids in correctly applying algebraic rules and performing operations like division and simplification.

Like Terms

Like terms are terms in an expression that have the same variable raised to the same power. In algebra and calculus, combining like terms is crucial for simplifying expressions efficiently.

• For instance, in the numerator of the exercise, breaking it into \[ (7 - 3x) + \frac{3}{2}x \], required combining constants with coefficients of like variable terms.
• This technique helps reduce expressions to their simplest form, which is especially helpful before applying any calculus rules or procedures.

Recognizing and combining like terms simplifies calculations and often makes broader problems easier to solve.