Problem 96 - Elementary and Intermediate Algebra

Question

Simplify each radical expression. Assume all variables are unrestricted. See Example 9. $$ \sqrt[5]{-32 x^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x\).

Step 1: Identify the Radical Expression

The problem provided is \( \sqrt[5]{-32 x^{5}} \). This expression is a fifth root of a product of a number and a variable expression.

Step 2: Simplify the Numerical Part

We first look at the numerical part: \(-32\). The fifth root of \(-32\) is \(-2\) because \((-2)^5 = -32\).

Step 3: Simplify the Variable Part

Next, simplify the expression with the variable. We have \(x^5\) inside the fifth root, which is \(\sqrt[5]{x^5} = x\), because raising a power to a root cancels out if the indices match.

Step 4: Combine the Results

Combine the simplified numerical and variable parts to obtain the final expression. Thus, \( \sqrt[5]{-32 x^{5}} = -2x \).

Key Concepts

Understanding the Fifth Root

When we talk about radical expressions, the term "fifth root" can come up quite often. Essentially, the fifth root finds the number that, when multiplied by itself five times, gives the original number back. It's similar to how a square root works, which involves multiplying the number by itself twice.
For example, if you take the fifth root of \(-32\), you are searching for a number that satisfies, \((-2)^5 = -32\). Indeed, \(-2\) is the number we need because multiplying \(-2\) by itself five times results in \(-32\).

• Fifth roots apply to both positive and negative numbers. A negative root indicates the number itself is negative.
• The index of the root (5 in this case) is crucial as it tells us how many times to multiply the root by itself.

This concept helps in breaking down and simplifying more complex expressions that involve exponents or roots.

The Importance of Simplification

Simplification in mathematics helps in reducing expressions to their most basic form. It makes calculations easier and results clearer. Simplification involves breaking down numbers or expressions carefully. You have to identify parts of the expression that can easily be calculated or canceled out.
In our provided radical expression, \(\sqrt[5]{-32 x^{5}}\), simplification follows two specific paths:

• Simplifying numbers, which involves determining the fifth root of \(-32\). This root came out as \(-2\).
• Simplifying variables, like with \(x^5\), allowing us to cancel the root and power since they match. This results in \(x\).

After simplifying both numerical and variable parts, we combine them to get a clean, final expression of \(-2x\). Simplifying this way helps us remove unnecessary complexity from mathematical expressions.

Working with Variable Expressions

Variable expressions involve not just numbers but also letters (variables like \(x, y\), etc.) that represent numbers. They are used often in radical expressions, which are expressions that contain roots.
When dealing with a variable in context of roots, remember these key points:

• The power of the variable inside the root, like \(x^5\), will often dictate the potential simplification. In our example, \(x^5\) becomes \(x\) because the root matches the power, cancelling each other out.
• Variables can be unrestricted, meaning they can hold any value permissible in real numbers. This implies that calculations assume variables could be positive, negative, zero, or any real number.

Understanding these principles of variable expressions makes it easier to manage and simplify them in real problems, helping bridge the gap between numbers and symbolic representations in algebraic expressions.