Problem 78 - Brief 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

\( \frac{u^2 v^2 w^2}{9} \)

Step 1: Apply the Power of a Power Rule

The expression is \( \frac{\left(u^3 v w^2\right)^2}{9(u^2 w)^2} \). We first apply the power of a power rule, \((a^m)^n = a^{m \cdot n}\), to the numerator and denominator separately. For the numerator, \((u^3 v w^2)^2 = u^{3 \cdot 2} v^{1 \cdot 2} w^{2 \cdot 2} = u^6 v^2 w^4\). For the denominator, \((u^2 w)^2 = u^{2 \cdot 2} w^{1 \cdot 2} = u^4 w^2\).

Step 2: Write the Simplified Expression

The expression after applying the power rule becomes: \( \frac{u^6 v^2 w^4}{9 u^4 w^2} \).

Step 3: Simplify the Fraction by Dividing Like Bases

Divide the like bases in the numerator and denominator. Starting with \( u^6 \) and \( u^4 \): \( \frac{u^6}{u^4} = u^{6-4} = u^2 \). For \( w^4 \) and \( w^2 \): \( \frac{w^4}{w^2} = w^{4-2} = w^2 \). The variable \( v \) has no corresponding base in the denominator, so it remains \( v^2 \).

Step 4: Final Simplified Expression

Combine the results of the simplification: \( \frac{u^2 v^2 w^2}{9} \).

Key Concepts

Power of a Power Rule

When working with exponents, you'll often come across the situation where you have a power of a power, like \((a^m)^n\). This is handled with the Power of a Power Rule. The rule states that you multiply the exponents together: \((a^m)^n = a^{m\cdot n}\). For example, if you see \((x^3)^2\), you multiply the exponents (3 and 2) to get \(x^6\). This rule makes the process of simplifying expressions with nested exponents straightforward and helps you to reduce the complexity of expressions.In our given exercise, we used this rule both in the numerator and the denominator: - Numerator: \((u^3 v w^2)^2 = u^{3 \cdot 2} v^{1 \cdot 2} w^{2 \cdot 2} = u^6 v^2 w^4\).- Denominator: \((u^2 w)^2 = u^{2 \cdot 2} w^{1 \cdot 2} = u^4 w^2\).This step is crucial in algebraic simplification as it sets the ground for further reduction of the expression.

Exponent Rules

Exponent rules are fundamental in algebra. They help us manipulate expressions involving powers efficiently. These rules are:

• Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of Powers Rule: \(\frac{a^m}{a^n} = a^{m-n}\), where \(a eq 0\)
• Power of a Power Rule: \((a^m)^n = a^{m\cdot n}\)
• Zero Exponent Rule: \(a^0 = 1\).

In the context of the expression we simplified, the quotient of powers rule was particularly important. For example:- Simplifying \(\frac{u^6}{u^4}\) resulted in \(u^{6-4} = u^2\).- Similarly, for \(w\), \(\frac{w^4}{w^2}\) became \(w^{4-2} = w^2\).Applying these rules allows us to combine and simplify terms into a less complicated expression, making complex algebra problems manageable.

Fraction Simplification

Simplifying fractions in algebra often involves reducing expressions by canceling shared factors between the numerator and the denominator. Here's how you approach it:- Identify common bases and use exponent rules like the Quotient of Powers Rule.- Factor out the greatest common factors if possible.In fraction simplification for our problem, the idea was to reduce the fractions to their simplest form by:- Beginning with the individual variable terms: for \(u^6/u^4\), subtract exponents to simplify to \(u^2\).- Next, handling other terms: \(w^4/w^2\) diminishes to \(w^2\).- Variables exclusive to the numerator like \(v^2\) stay unchanged as there's no corresponding term in the denominator.Fraction simplification is central to algebra because it streamlines expressions, making them easier to evaluate and understand. After completing this simplification, the expression appears much tidier as \(\frac{u^2 v^2 w^2}{9}\). By consistently applying these principles, one can handle increasingly complex algebraic operations with confidence.