Problem 39 - Algebra 2

Question

Simplify each expression. $$ w^{-\frac{4}{5}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{1}{w^{\frac{4}{5}}} \).

Step 1: Understand the Problem

We are asked to simplify the expression \( w^{-\frac{4}{5}} \). This involves rewriting the expression in a simpler or more standard form by using exponent rules.

Step 2: Apply the Rule for Negative Exponents

Recall the rule for negative exponents, which states that \( a^{-b} = \frac{1}{a^b} \). We apply this to the expression \( w^{-\frac{4}{5}} \) to get \( \frac{1}{w^{\frac{4}{5}}} \).

Step 3: Simplify the Expression Further (if necessary)

The expression \( \frac{1}{w^{\frac{4}{5}}} \) is already in a simplified form. However, it could be rewritten using a radical if desired: \( \frac{1}{\sqrt[5]{w^4}} \). In this case, we'll leave it in exponential form.

Key Concepts

Simplifying Expressions

Simplifying expressions involves rewriting terms to make them more understandable and manageable. When working with exponents, the goal is to express the term in the simplest form by applying relevant mathematical rules. For example, in our exercise with the expression \( w^{-\frac{4}{5}} \), simplifying it requires an understanding of how negative and fractional exponents function. By using established exponent rules, such as converting negative exponents into fractions, we can transform complex terms into a more standard and accessible form.
Careful manipulation and simplification can assist in making calculations easier, and this is particularly useful in algebra when expressions become part of larger equations to solve.

Exponent Rules

Exponent rules are fundamental principles that define how to manipulate expressions involving powers. These rules simplify the process of dealing with powers by providing shortcuts and strategies for rewriting expressions. One important exponent rule is the negative exponent rule, which states:

• \( a^{-b} = \frac{1}{a^b} \)

If you have a term with a negative exponent, like \( w^{-\frac{4}{5}} \), you can rewrite it as \( \frac{1}{w^{\frac{4}{5}}} \). This conversion allows us to work with positive exponents, which are typically more straightforward to handle.
By applying these rules systematically, complicated expressions become approachable and comprehensible, facilitating easier calculation and further manipulation.

Fractional Exponents

Fractional exponents represent roots in a compact notation, and understanding their implications is key to simplifying expressions correctly. A fractional exponent such as \( \frac{4}{5} \) in \( w^{\frac{4}{5}} \) indicates a root as well as a power:

• The denominator \( 5 \) specifies a fifth root.
• The numerator \( 4 \) signifies the power that the base is raised to before or after taking the root.

So, \( w^{\frac{4}{5}} \) is equivalent to \( \sqrt[5]{w^4} \). Moreover, rewriting expressions with fractional exponents in radical form can often make it easier to understand certain operations, like simplification or division.
It's crucial to recognize that converting between these representations does not change the value of the expression but merely offers a different way of looking at the problem.