Problem 47 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[5]{-y^{20}} $$

Step-by-Step Solution

Answer

The simplified expression of \(\sqrt[5]{-y^{20}}\) is \(y^4\).

Step 1: Identify the root and the power

The given expression is \(\sqrt[5]{-y^{20}}\). Here, the fifth root needs to be taken. The number under the root is \(-y^{20}\). The power of y is an even number.

Step 2: Remove power from under the root

As y is raised to an even power, we can take it out of the root. This results in \(-y^4\).

Step 3: Apply the absolute value when needed

Since \(-y^4\) is necessarily a negative number, it cannot be simplified further without considering absolute value. Therefore, \(-y^4\) turns into \(|-y^4|\). The absolute value of a negative number is a positive number, so \(|-y^4|\) turns into \(y^4\).

Key Concepts

Understanding Absolute Value

Absolute value is a mathematical concept used to describe the magnitude of a number, regardless of its sign. When dealing with expressions, especially involving roots and powers, you often encounter negative numbers inside the root. Here, absolute values become crucial.
- Absolute value is denoted by vertical bars, such as \(|x|\).
- For any real number, the absolute value simply reflects its non-negative magnitude.
- For example, the absolute value of -7 is 7 because it measures the distance from zero on the number line.In the context of radical expressions, certain power distributions can result in negative numbers within roots. It's important to ensure we use absolute value symbols to reflect the non-negative nature of the resulting value after simplification. In our example, the expression simplifies to \(|-y^4|\), which then becomes \(|y^4|\). By rule, this is equal to \(|y^4|\), thus ensuring our expression remains within the realm of real numbers.

Exploring Roots and Powers

When dealing with expressions involving roots and powers, understanding their relationship is crucial. Roots are considered the inverse operation of powers. For example, a square root is the inverse operation of squaring a number.
- In general, the n-th root of a number shares an inverse relationship with raising to the power of n.
- To express this mathematically: if \(x^n = y\), then \((x = \sqrt[n]{y})\).In many cases, roots and powers are used together to simplify expressions format. For instance, in the expression \(\sqrt[5]{-y^{20}}\), the power of y is 20, which is reduced by a factor that matches the root (in our case, the fifth root). The result when solving is \(-y^4\), and then addressing the absolute value as needed, as roots and powers both affect whether the expression inside a root results in a positive or negative outcome.

Simplifying Expressions Step by Step

Simplifying radical expressions often involves a systematic approach. Let's break it down:

• Identify the root and power: Recognize the base and the exponent under your radical sign. This helps set up how you will simplify the expression.
• Calculate the root: This involves the inverse operation of raising to a power, meaning if you have \(\sqrt[n]{y^m}\), you will emit the power to simplify it to \(y^{m/n}\).
• Check for negative results: If your root results in a negative expression, use absolute value to maintain the positivity of your expression, ensuring the mathematical validity.
• Write your answer cleanly: Once absolute value has been considered, your simplified expression should reflect a solution free from ambiguity.

In our case, following these steps aided us in simplifying \(\sqrt[5]{-y^{20}}\) into \(|y^4|\), emphasizing how methodically progressing through calculations ensures proper expression simplification.