Problem 77 - Brief 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, let's expand the expression in the numerator. The numerator is \((2u^{2}vw^{3})^{2}\). To expand it, apply the power to each component:\[(2u^2vw^3)^2 = (2)^2(u^2)^2(v)^2(w^3)^2 = 4u^4v^2w^6.\]

Step 2: Expand the Denominator

Now, let's handle the denominator, which is \(4(u w^{2})^{2}\). Here, apply the power to each term inside the parentheses first:\[(u w^2)^2 = u^2 w^4.\]Thus, the expression becomes:\[4u^2w^4.\]

Step 3: Combine and Simplify

Combine the expanded expressions in the fraction:\[\frac{4u^4v^2w^6}{4u^2w^4}.\]Cancel out the common factors in the numerator and the denominator:- The \(4\) in both the numerator and the denominator cancel out.- The \(u^2\) in the numerator and denominator reduces \(u^4\) to \(u^2\) in the numerator.- The \(w^4\) in the denominator cancels with part of \(w^6\) in the numerator, leaving \(w^2\) in the numerator.This simplifies the expression to:\[u^2v^2w^2.\]

Step 4: Final Simplified Expression

The fully simplified expression, after combining and cancelling the terms, becomes:\[u^2v^2w^2.\] This is the final answer to the simplification problem.

Key Concepts

Exponents and Powers

Exponents and powers are fundamental concepts in algebra that allow us to express large numbers or repeated multiplications in a concise way. Exponents, also known as powers, tell us how many times to multiply a base by itself. For example, in the term \(2^3\), the number 2 is the base, and 3 is the exponent, which means we multiply 2 by itself three times:

• \(2^3 = 2 \times 2 \times 2 = 8\)

When dealing with algebraic expressions like \((2u^2vw^3)^2\), we apply the power to each factor within the parentheses. The power of 2 means that each component inside is multiplied by itself, resulting in:

• \((2)^2 = 4\)
• \((u^2)^2 = u^4\)
• \((v)^2 = v^2\)
• \((w^3)^2 = w^6\)

Exponents follow specific rules which simplify calculations significantly. Having a solid grasp of these basic rules makes manipulating expressions and solving equations much easier.

Numerator and Denominator

In any fraction, the top part is known as the numerator, and the bottom part is the denominator.Understanding how each component interacts is essential for simplification. In expressions like \(\frac{(2u^2vw^3)^2}{4(uw^2)^2}\), the numerator and denominator have their roles:

• The numerator \((2u^2vw^3)^2\) represents the value to be divided.
• The denominator, such as \(4(uw^2)^2\), signifies how many parts the numerator is split into.

To simplify fractions, each part is expanded separately to facilitate easy comparison and cancellation of similar components. Knowing how to properly expand and rearrange both parts is necessary for engaging in processes like factor cancellation, ultimately reducing expressions to their simplest form.

Factor Cancellation

Factor cancellation involves cancelling common factors that appear in both the numerator and the denominator of a fraction. It is a key step in simplifying algebraic expressions. Let's look at the fraction from the exercise: \(\frac{4u^4v^2w^6}{4u^2w^4}\).Start by identifying common factors. Here, both the numerator and the denominator have the following:

• The factor \(4\) cancels out as it appears in both.
• For \(u\): \(u^4\) in the numerator and \(u^2\) in the denominator allow us to reduce \(u^4\) to \(u^2\) in the numerator.
• For \(w\): \(w^6\) in the numerator and \(w^4\) in the denominator enable reduction to \(w^2\) in the numerator.

Upon cancelling these factors, the simplified result is \(u^2v^2w^2\). Factor cancellation is powerful because it reduces expressions to simpler forms that are often more manageable and easier to understand in subsequent calculations or applications.