Problem 80 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( w^3 \).

Step 1: Simplify the Numerator

First, simplify the expression inside the parentheses in the numerator: \( (ww^3)^2 \). This expression involves multiplying \( w \) and \( w^3 \), giving us \( w^{1+3} = w^4 \). Raising \( w^4 \) to the power of 2 gives us \( (w^4)^2 = w^{4 imes 2} = w^8 \). The simplified numerator is \( w^8 \).

Step 2: Simplify the Denominator

Next, simplify the denominator \( w^3w^2 \). Since both terms are bases \( w \), you can add the exponents according to the rule of multiplying powers: \( w^{3+2} = w^5 \). Therefore, the simplified denominator is \( w^5 \).

Step 3: Divide the Simplified Expressions

Now divide the simplified numerator \( w^8 \) by \( w^5 \) in the denominator using the rule of dividing powers (subtract the exponents): \( \frac{w^8}{w^5} = w^{8-5} = w^3 \). Thus, the expression simplifies to \( w^3 \).

Key Concepts

Exponentiation Rules

Exponentiation involves raising a number to the power of an exponent, which means multiplying the number by itself a certain number of times. Understanding the basic rules of exponents is key to simplifying expressions.

• Product of Powers: When you multiply two powers that have the same base, simply add the exponents together. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: If you have a power raised to another power, multiply the exponents. This means \((a^m)^n = a^{m \times n}\).
• Power of a Product: When you have a power that applies to a product, such as \((ab)^n\), apply the exponent to each factor: \(a^n \cdot b^n\).

These rules provide the foundation for simplifying more complex expressions that involve multiple exponentiations like \((ww^3)^2\). This expression can be reduced using both the multiplication of powers and power of a power rules.

Multiplying Powers

Multiplying powers is about using exponents to combine like bases. When you multiply numbers or variables with the same base, you will add their exponents. This technique helps in reducing complex expressions.Let's consider the expression \( (ww^3)^2 \). Here, inside the parenthesis, we first multiply \(w \times w^3\). With the rule of multiplying powers (add the exponents), this simplifies to \(w^{1+3} = w^4\). Therefore, we can transform \((ww^3)^2\) into \((w^4)^2\).Once we have \(w^4\), we apply the power of a power rule, raising it to the second power: \((w^4)^2 = w^{4 \times 2} = w^8\). By consistently using these methods, you can make any algebraic expression with powers clearer and easier to handle.

Dividing Powers

Dividing powers involves subtracting exponents, which is crucial when simplifying expressions where the same base appears in both the numerator and denominator.In our example, after simplifying \((ww^3)^2\) to \(w^8\), and the denominator \(w^3w^2\) to \(w^5\), we move to divide the two:

• According to the division rule of exponents, when dividing like bases, subtract the exponents of the denominator from the numerator: \(\frac{w^8}{w^5} = w^{8-5} = w^3\).

This method allows you to simplify expressions quickly by reducing them into more manageable forms. With practice, working with these rules becomes intuitive and helps with tackling various algebra problems with ease.