Problem 35 - Algebra A Combined Function

Question

Find each root. Assume that all variables represent nonnegative real numbers. $$ -\sqrt[4]{16} $$

Step-by-Step Solution

Answer

The root is -2.

Step 1: Understand the Root

The expression \(-\sqrt[4]{16}\) involves finding the fourth root of 16. The root denominator 4 indicates we are looking for a number that, when raised to the power of 4, equals 16.

Step 2: Simplify the Root

The fourth root of 16 can be expressed as \(16^{1/4}\). To simplify \(16^{1/4}\), we can factor 16: it is \(2^4\). Therefore, \((2^4)^{1/4} = 2^{(4/4)} = 2\).

Step 3: Apply the Negative Sign

After simplifying, we found that the fourth root of 16 is 2. However, the expression has a negative sign in front of the root, so \(-\sqrt[4]{16} = -2\).

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that contain roots, such as the square root, cube root, or any higher-order roots like the fourth root. These roots are represented with a radical sign (√) and may include an index indicating the degree of the root. For instance,

• The radical expression \( \sqrt[4]{16} \) is a fourth root.
• In radical terms, the number underneath the radical sign is known as the radicand, which in this case is 16.

To evaluate radical expressions, we need to understand what root or power simplifies the radicand back to its base number. In this exercise, the fourth root of 16 is found using the factorization of 16 and simplifying it to one of its base factors. It's crucial to identify whether the radical expression involves only positive numbers or if there are additional considerations such as negative signs, which impact the final outcome.

Simplifying Roots

Simplifying roots is about breaking down radical expressions to their simplest form. Let's take the example \( \sqrt[4]{16} \).

• First, recognize the type of root: the index here is 4, indicating a fourth root.
• Identify if the radicand can be expressed as a power of another number, like in this case \(16 = 2^4 \).
• Applying properties of exponents, \( (2^4)^{1/4} = 2^{4/4} = 2 \), simplifying the fourth root of 16 to 2.

Simplifying roots often involves these power and factor techniques, reducing the expression to the most basic form feasible. One must be careful with simplification to correctly apply power rules and achieve proper expression without altering the problem's context or intent.

Negative Signs in Roots

Negative signs in front of roots mean more than just flipping a positive answer to negative. They are crucial in interpreting the solution accurately. In the exercise, the question is \( -\sqrt[4]{16} \).

• First compute the root: \(\sqrt[4]{16} = 2\).
• Then, apply the negative sign: the expression becomes \(-2\).

It's essential to distinguish between a negative sign outside the radical and potential negative values inside; this affects the roots in complex ways, especially with even roots which discard negative results under the radicand ensuring real-number outcomes. Therefore, always keep these negative signs in mind when simplifying and concluding any root to guarantee accuracy.