Problem 85 - Applied Calculus

Question

Simplify. $$ \frac{\left(2 u^{2} v w^{3}\right)^{2}}{4\left(u w^{2}\right)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( u^2v^2w^2 \).

Step 1: Expand the Numerator

First, expand the expression in the numerator: \[ (2u^2 vw^3)^2 = (2)^2 (u^2)^2 (v)^2 (w^3)^2 \]This simplifies to:\[ 4u^4v^2w^6 \]

Step 2: Expand the Denominator

Next, expand the expression in the denominator:\[ 4(u w^2)^2 = 4(u^2)(w^2)^2 \]This simplifies to:\[ 4u^2w^4 \]

Step 3: Simplify the Fraction

Now, divide the expanded numerator by the expanded denominator:\[ \frac{4u^4v^2w^6}{4u^2w^4} \]Cancel the common factor of 4 from both numerator and denominator, and simplify:\[ \frac{u^4v^2w^6}{u^2w^4} \]

Step 4: Simplify the Exponents

Further simplify by subtracting the exponents of the common bases:- For \( u \): \( u^{4-2} = u^2 \)- For \( w \): \( w^{6-4} = w^2 \)The expression simplifies to:\[ u^2v^2w^2 \]

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to the power of another number. In simpler terms, it involves multiplying a number by itself a specified number of times.
For instance, when we see something like \( x^n \), this means that we multiply \( x \) by itself \( n \) times.

Understanding how to work with exponentiation is crucial, especially in terms of simplification as seen in the original exercise.

• Power of a Product: When you have something like \((ab)^n\), you apply the exponent to each factor inside the parentheses. That means \((ab)^n = a^n b^n\).
• Power of a Power: When you have an exponent raised to another exponent, you multiply them. For example, \((x^m)^n = x^{m \cdot n}\).

In simplifying the expression \((2u^2 vw^3)^2\) from the original exercise, each component inside the parentheses is raised to the power of 2. This results in \((2)^2 (u^2)^2 (v)^2 (w^3)^2\), demonstrating the multiplication of exponents.
By expanding through exponentiation, we simplify the components individually to ultimately achieve the numerator as \( 4u^4v^2w^6\).

Rational Expressions

A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Simplifying rational expressions means making the fraction as simple as possible by factoring and reducing.In order to simplify rational expressions, you should:

• Factor Completely: Factor the numerator and the denominator as much as possible.
• Identify Common Factors: Once factored, look for common factors in the numerator and the denominator.
• Cancel Common Factors: Divide both the numerator and the denominator by the common factors.

In the original exercise, the rational expression \( \frac{4u^4v^2w^6}{4u^2w^4}\) is simplified by canceling out the common factor of 4 initially, yielding \(\frac{u^4v^2w^6}{u^2w^4}\).
This exercise highlights the importance of recognizing common factors, allowing the simplification to \( u^2v^2w^2 \) after simplifying the exponents further.

Polynomial Division

Polynomial division is a process that involves dividing a polynomial by another polynomial. This is comparable to long division for numbers but involves algebraic expressions. In the context of algebraic simplification, polynomial division helps in breaking down more complex fractions into their simplest forms by dealing with powers and simplifying across terms.

• Similar Bases: In division, when dealing with similar bases, you subtract the exponent in the denominator from the exponent in the numerator (\(x^a / x^b = x^{a-b}\)).
• Cancellation of Terms: Once you simplify exponents, you can cancel out the terms that are common in both parts of the fraction.

The polynomial division here involves subtracting exponents in the expression \( \frac{u^4v^2w^6}{u^2w^4} \).
By "dividing" \( u^4 \) by \( u^2 \) and \( w^6 \) by \( w^4 \), you get \( u^2 \) and \( w^2 \) respectively, showcasing how polynomial division simplifies the expression to \( u^2v^2w^2 \). This exercise emphasizes the subtraction of exponents and the reduction of complex expressions through polynomial division.