Problem 114 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt[6]{(x+4)^{6}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x+4 \).

Step 1: Identify the Expression

The given expression is \( \sqrt[6]{(x+4)^{6}} \), which is a sixth root of \( (x+4)^6 \). Our task is to simplify this expression.

Step 2: Recognize the Power and Root Relationship

Notice that the power of the expression inside the radical (6) matches the root (also 6). This means the two operations can cancel each other out. Specifically, \( \sqrt[n]{a^n} = a \) for any real number \( a \) and integer \( n \).

Step 3: Simplify the Radical

Apply the power and root cancellation property: \( \sqrt[6]{(x+4)^{6}} = (x+4) \). The sixth root of a sixth power returns the base itself.

Key Concepts

Power and Root Relationship

When dealing with radical expressions, especially when you have a situation like \( \sqrt[n]{a^n} \), the relationship between power and root is crucial. Essentially, if you have a number raised to a power that is the same as the root, they cancel each other out. A simpler way to understand this is through an example.

Imagine you have \( \sqrt[6]{(x+4)^{6}} \). Here, the expression inside the root is raised to the sixth power. The root itself is the sixth root. The two operations—raising to a power and taking a root—"undo" each other, leaving us with just the base, \( x+4 \).

This cancellation property arises from the basic principle that raising a number to an exponent and then taking a root of the same degree results in the base number. It's a neat trick to simplify expressions quickly. Remember: \( \sqrt[n]{a^n} = a \) for any real number \( a \) and integer \( n \).

Radical Simplification

Simplifying radical expressions often involves making them as neat and concise as possible. Recognizing patterns, such as the power and root relationship, is a big part of this process. Simplification is all about reducing the expression to its simplest form while maintaining its value.

Take the expression \( \sqrt[6]{(x+4)^{6}} \) as an example. At first glance, it seems complex due to the radical and the exponent. However, by applying the power and root relationship, it immediately simplifies to \( x+4 \). This transformation helps to streamline calculations and make the problem more manageable.

To simplify effectively, it's important to:

• Identify any perfect power inside the radical.
• Apply properties of exponents, such as dividing exponents when taking a root.
• Use properties such as \( \sqrt[n]{a^n} \).

These strategies allow radical expressions to be simplified easily and efficiently.

Expression Properties

Understanding the properties of expressions, especially in algebra, is essential for simplifying them. Radical expressions come with their own set of properties that help you handle them more effectively.

Some key properties include:

• Non-Negativity: When dealing with real numbers, the expression inside a radical must be non-negative for even roots to produce a real number output.
• Maintaining Order: Operations like addition and multiplication are preserved within radicands. For example, \( \sqrt[a]{bc} = \sqrt[a]{b} \times \sqrt[a]{c} \).
• Radical to Rational: An expression under a radical of the form \( \sqrt[n]{a^m} \) can be rewritten as \( a^{m/n} \).

These properties aid in both recognizing patterns quickly and reducing an expression to its simplest form. By understanding these rules, you can make significant progress in working with and simplifying radical expressions.