Problem 72 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Simplify the Numerator

Start by simplifying the expression in the numerator, \( (w w^3)^2 \). First notice that \( w w^3 = w^1 imes w^3 = w^{1+3} = w^4 \). Then, raise this result to the power of 2: \( (w^4)^2 = w^{4 imes 2} = w^8 \).

Step 2: Simplify the Denominator

Next, simplify the expression in the denominator, \( w^3 w^2 \). Use the property of exponents that states \( a^m imes a^n = a^{m+n} \), which gives \( w^3 imes w^2 = w^{3+2} = w^5 \).

Step 3: Divide the Simplified Terms

Now divide the simplified numerator by the simplified denominator. Use the division rule of exponents: \( \frac{w^m}{w^n} = w^{m-n} \). In this case, \( \frac{w^8}{w^5} = w^{8-5} = w^3 \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, especially when dealing with exponents. It involves reducing an expression to its simplest form while maintaining its value. In the given exercise, we start by looking at the expression in the numerator: \( (w w^3)^2 \). Understanding how to simplify requires breaking down each term:

• Recognize that \( w w^3 \) can be rewritten by combining like bases: \( w^1 \cdot w^3 = w^{1+3} = w^4 \).
• Then, raise \( w^4 \) to the power of 2: \( (w^4)^2 = w^{4 \cdot 2} = w^8 \).

Breaking down each step like this helps in grasping the logic behind the operations. By simplifying step-by-step, you can handle more complex expressions with confidence. When tackling similar problems, always look for ways to break down exponents before applying operations.

Properties of Exponents

Exponents hold unique properties that make calculations more efficient. Knowing these can save time and reduce errors in your work. Here are some crucial properties to keep in mind when simplifying expressions:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\). Combining exponents when multiplying like bases, as seen in both the numerator and denominator simplification steps, such as \(w^4 \to w^8\) and \(w^3 \cdot w^2 \to w^5\).
• Power of a Power: \((a^m)^n = a^{m \cdot n}\). This helps evaluate expressions like \( (w^4)^2 = w^8 \).

Understanding these rules not only aids in simplifying expressions but also enhances your problem-solving toolkit. Always remember that the base remains the same, while the exponents are manipulated according to these properties.

Division of Powers

When you have quotients with identical bases, division of powers becomes an essential tool. This concept allows you to simplify expressions by subtracting exponents of like bases.In the simplification process:

• Use the rule \( \frac{a^m}{a^n} = a^{m-n} \) to divide powers of the same base.

Applying this rule to our problem, the division of \( \frac{w^8}{w^5} \) gives \( w^{8-5} = w^3 \). The division step reduces a more complex expression into a simple result.Remember, division of powers is only straightforward when the bases are the same. Always ensure that you correctly apply the rule to get to the simplest possible form of the expression. This method not only simplifies the calculation but also reinforces understanding of the fundamental exponent rules.