Problem 79 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(w^5\).

Step 1: Simplify the Numerator

First, let's simplify the numerator \((w \cdot w^2)^3\). Apply the property of exponents \((a^m)^n = a^{m \cdot n}\). Here, \(w \cdot w^2 = w^1 \cdot w^2 = w^{1+2} = w^3\). Thus, \((w^3)^3 = w^{3 \cdot 3} = w^9\).

Step 2: Simplify the Denominator

Now, simplify the denominator \(w^3 \cdot w\). Use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\). Therefore, \(w^3 \cdot w = w^{3+1} = w^4\).

Step 3: Divide the Powers of w

Use the quotient property of exponents that says \(\frac{a^m}{a^n} = a^{m-n}\) to simplify this expression. Therefore, \(\frac{w^9}{w^4} = w^{9-4} = w^5\).

Key Concepts

Exponents Rules

When tackling algebraic expressions with exponents, it is crucial to understand the fundamental rules governing them. One key rule is the "power of a power" property, which states that if you have a base raised to an exponent, and then the whole is raised to another exponent, you multiply the exponents:

• \((a^m)^n = a^{m \cdot n}\).

This means you take the base exponent and multiply it by the outer exponent. Another important rule is the "product of powers" property, which helps in simplifying expressions where exponents are involved in multiplication:

• \(a^m \cdot a^n = a^{m+n}\).

By applying these rules, calculating powers and simplifying expressions becomes much more manageable. Understanding these basic concepts enables you to work through expressions step by step, increasing accuracy and confidence.

Numerator and Denominator Simplification

In algebra, simplifying the numerator and denominator of a fraction is a crucial step. This involves applying the exponents rules mentioned earlier. Let's consider the given numerator \((w \cdot w^2)^3\).

• First, combine the inner powers: \(w^1 \cdot w^2 = w^{1 + 2} = w^3\).
• Then, raise this result to the third power using the power of a power rule: \((w^3)^3 = w^{3 \cdot 3} = w^9\).

Eventually, you get a simplified expression in the numerator. On the other hand, for the denominator like \(w^3 \cdot w\), use the product of powers rule:

• Combine the exponents: \(w^3 \cdot w^1 = w^{3 + 1} = w^4\).

Simplifying both parts separately before combining them makes it easier to handle complex fractions in algebraic expressions.

Quotient Property of Exponents

The quotient property of exponents is a helpful tool for dividing expressions that include the same bases. This rule simplifies an expression of the form \(\frac{a^m}{a^n}\) by subtracting the exponents:

• \(\frac{a^m}{a^n} = a^{m-n}\).

Applying this property aids in simplifying expressions where you have divided terms with the same base. For instance, consider \(\frac{w^9}{w^4}\).

• Subtract the exponent of the denominator from the exponent of the numerator: \(w^{9-4} = w^5\).

Understanding and practicing this property allows you to tackle fractional exponents confidently and fluently. It simplifies complex fractions, making them more manageable and enhancing overall algebraic understanding.