Problem 86 - Applied Calculus

Question

Simplify. $$ \frac{\left(u^{3} v w^{2}\right)^{2}}{9\left(u^{2} w\right)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{u^2 v^2 w^2}{9}\).

Step 1: Expand the Numerator

First, expand the numerator \((u^3 v w^2)^2\). To do this, apply the power rule \((a^m b^n)^p = a^{mp} b^{np}\): \((u^3 v w^2)^2 = u^{6} v^2 w^4\).

Step 2: Expand the Denominator

Now, expand the denominator \(9(u^2 w)^2\). Apply the power rule and multiplication to get: \(9(u^2 w)^2 = 9 imes u^4 imes w^2 = 9u^4w^2\).

Step 3: Simplify the Expression

Combine the expanded numerator with the denominator: \[\frac{u^6 v^2 w^4}{9 u^4 w^2}\]. Now simplify by dividing each term in the numerator by the corresponding term in the denominator: \[u^{6-4} v^2 w^{4-2} = \frac{u^2 v^2 w^2}{9}\].

Step 4: Final Simplification

Ensure that the expression is fully simplified. After confirming, the simplified expression is \[\frac{u^2 v^2 w^2}{9}\].

Key Concepts

Exponent Rules

Exponent rules are essential tools in algebra, enabling us to manage powers efficiently. In the exercise, we use a key rule:

• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)

This rule helps us simplify expressions by distributing the external exponent across each of the factors inside the parentheses. Let's look at how these rules apply when simplifying algebraic expressions:
First, expand the expression \( (u^3 v w^2)^2 \). Apply the \'power of a power\' rule to each part inside:

• \( u^3 \rightarrow (u^3)^2 = u^{3\cdot2} = u^6 \)
• \( v \rightarrow (v)^2 = v^2 \)
• \( w^2 \rightarrow (w^2)^2 = w^{2\cdot2} = w^4 \)

The expanded form for the numerator becomes \( u^6 v^2 w^4 \). Recognizing and correctly applying exponent rules leads to accurate and simplified expressions, every step of the way.

Numerator and Denominator

In any fraction, the numerator sits on top and the denominator on the bottom. Understanding this position is crucial for proper algebraic manipulation. Let's explore how we work with the numerator and denominator in this exercise.
Starting with the numerator \( (u^3 v w^2)^2 \), we expanded it using exponent rules to \( u^6 v^2 w^4 \). This is placed over the denominator \( 9(u^2 w)^2 \).
Now, to simplify the denominator, we expand and simplify:

• \( 9 \rightarrow \text{stays the same as it\'s a constant} \)
• \( (u^2)^2 = u^{2\cdot2} = u^4 \)
• \( (w)^2 = w^2 \)

This expansion leads to the denominator becoming \( 9u^4w^2 \). With both the numerator and the denominator clearly expanded, we proceed to the next step of simplification.

Algebraic Manipulation

Algebraic manipulation is the process of reshaping algebraic expressions into more manageable forms. In this exercise, one key action was reducing the fraction.
We began with the fraction \[\frac{u^6 v^2 w^4}{9 u^4 w^2}\]. The goal is to cancel common factors from the numerator and denominator, making the expression simpler. Here's how:

• Divide \( u^6 \) by \( u^4 \) to get \( u^{6-4} = u^2 \)
• \( v^2 \) stays unchanged since \( v \) is not in the denominator
• Divide \( w^4 \) by \( w^2 \) to get \( w^{4-2} = w^2 \)

Thus, after careful reduction of like terms, the expression simplifies to \[\frac{u^2 v^2 w^2}{9}\].
Algebraic manipulation, especially factor cancellation, transforms expressions into their simplest forms, making them easier to interpret and use, for future calculations or solving equations.