Problem 48 - Intermediate Algebra

Question

Simplify. Assume that the variables represent any real number. $$ \sqrt[4]{16 x^{4}} $$

Step-by-Step Solution

Answer

The simplified form is \(2x\).

Step 1: Identify the Expression

The expression given is \( \sqrt[4]{16 x^{4}} \). This means we need to find the fourth root of \( 16x^4 \).

Step 2: Simplify the Radicand

Notice that \(16\) is a perfect fourth power since \(16 = 2^4\). Similarly, \(x^4\) is also \((x^1)^4\). This allows us to rewrite the expression inside the root as \((2x)^4\).

Step 3: Apply the Fourth Root

The fourth root of \((2x)^4\) is simply \(2x\). This is because the fourth root and the fourth power cancel each other out.

Step 4: Write the Final Simplified Expression

The simplified form of the original expression \( \sqrt[4]{16 x^{4}} \) is \(2x\).

Key Concepts

Understanding the Fourth Root

When you take the fourth root of a number, you're looking for a value that, when multiplied by itself four times, equals that number. This can be written using a radical sign with a little number 4 on the upper left: \( \sqrt[4]{a} \), where \( a \) is the number you want the fourth root of. This process is akin to identifying an 'invisible' exponent that will undo a fourth power operation. Think about it like this:

• If \( b^4 = a \), then \( \sqrt[4]{a} = b \).

Thus, the fourth root and possibly the most familiar example involves perfect fourth powers like 16, because \( 2^4 = 16 \). By understanding fourth roots, you are essentially reversing the process of raising something to the fourth power.

Real Numbers and Their Role

Real numbers include pretty much any number you can think of, except for imaginary numbers, like \( \sqrt{-1} \). They encompass all positive and negative whole numbers, fractions, and decimals that you use daily. This concept also includes irrational numbers like \( \sqrt{2} \) that cannot be expressed as a simple fraction. In the context of radicals, real numbers are important because they define the set of values that variables can take.

• For example, when our expression contains a variable \( x \), we are assuming \( x \) can be any real number.

This assumption ensures that our simplified expression remains applicable and correct within the realm of real number operations.

Perfect Fourth Power Basics

Perfect fourth powers are numbers that can be written as another number raised to the fourth power. In other words, if \( a = b^4 \), then \( a \) is a perfect fourth power. This includes numbers like 1, 16, and 81, which can be expressed as \( 1^4, 2^4, \) and \( 3^4 \) respectively. Recognizing them is crucial in simplifying radicals because:

• They allow the radical to be easily undone – transforming a number from its complex form back to its base.
• For example in our given problem, recognizing 16 as \( 2^4 \) makes simplification straightforward.

With practice, identifying perfect powers becomes second nature, streamlining the process of simplifying complex radical expressions.