Problem 43 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[4]{16 c^{4}} $$

Step-by-Step Solution

Answer

The simplified expression of \(\sqrt[4]{16 c^{4}}\) is \(2|c|\).

Step 1: Apply the rule to numerical term

Find the fourth root of the number \(16\). The fourth root of \(16\) is \(2\), because when \(2\) is raised to the power of \(4\) (\(2^4\)), it equals 16.

Step 2: Apply the rule to the variable

For \(c^{4}\) raised to the \(1/4\) (fourth root), apply the property \( \sqrt[n]{a^n} = a \). Therefore, \(\sqrt[4]{c^{4}}\) simplifies to \(|c|\), as we take the absolute value to ensure that we only consider the real, positive root.

Step 3: Combine the results

Now we combine the results from Step 1 and Step 2. The simplified form of \(\sqrt[4]{16c^{4}}\ is \(2|c|\)

Key Concepts

Fourth Roots

In mathematics, finding the fourth root of a number means identifying a value which, when raised to the power of 4, equals the original number. For example, to find the fourth root of 16, we need to determine what number multiplied by itself four times equals 16.

• The fourth root of 16 is 2.
• This is because \(2^4 = 2 \times 2 \times 2 \times 2 = 16\).

When working with fourth roots, you're essentially reversing the exponentiation process. If a number inside the radical can be broken down into repeated factors, like 16 being \(2^4\), you can easily find its fourth root.
This idea is also applied to variables, so \(\sqrt[4]{c^4} = |c|\), which brings us to the role of the absolute value in this process.

Absolute Value

Absolute value represents the non-negative value of a number, essentially ignoring any negative sign. It is denoted by two vertical bars around the number or expression, like \(|c|\).
When you see \(|c|\), it's asking for the distance of \(c\) from zero on the number line without considering direction. This is crucial when simplifying radicals, such as \(\sqrt[4]{c^4}\).

• Usually, we require absolute values to ensure all roots are non-negative. This is because when dealing with even roots like square roots or fourth roots, the principal (main) root is always non-negative.
• So, in the expression \(\sqrt[4]{c^4}\), we result in \(|c|\) to maintain its non-negativity.

This ensures that our result is mathematically correct and acknowledges all possible values of \(c\), whether \(c\) starts as positive or negative.

Algebraic Expressions

An algebraic expression involves numbers, variables, and arithmetic operations. Simplifying these expressions means rewriting them in a more concise form while keeping their values unchanged.
In the exercise \(\sqrt[4]{16c^4}\), you're dealing with both numbers (16) and variables (\(c^4\)).

• The objective is to simplify each aspect individually using mathematical rules such as roots and powers.
• The result combines known formulas, like \(a^4\) rooted to pure numbers or absolute values, simplifying to simpler terms.

In this case, you convert \(16\) into \(2\), and \(c^4\) into \(|c|\), resulting in a streamlined expression: \(2|c|\).
Algebraic simplification is important because it makes working with expressions easier, paving the way for more complex calculations or further simplifications.