Problem 29 - Intermediate Algebra

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: Identify the Root Type

The expression involves the fourth root, which is denoted by \( \sqrt[4]{} \). We need to find the fourth root of 16.

Step 2: Understand Fourth Roots

The fourth root of a number \( a \) is a value \( b \) such that \( b^4 = a \). We must find a number \( b \) fulfilling \( b^4 = 16 \).

Step 3: Calculate Fourth Root of 16

Determine the number whose fourth power equals 16. Since \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \), we find that \( \sqrt[4]{16} = 2 \).

Step 4: Account for the Negative Sign

The expression contains a negative sign in front of \( \sqrt[4]{16} \). Hence, the result is the negative of the fourth root of 16. Therefore, \( -\sqrt[4]{16} = -2 \).

Step 5: Verify the Solution

Verify by calculating if \( (-2)^4 = 16 \). Since \( (-2) \times (-2) \times (-2) \times (-2) = 16 \), the calculation checks out.

Key Concepts

Negative Numbers in Algebra

Negative numbers can sometimes be tricky in algebra, especially when they're combined with operations like roots and powers.

• Understanding the Negative Sign: In the problem, the negative sign in front of the fourth root indicates that after calculating the root, the value should be negated. For example, in \(-\sqrt[4]{16}\), once you find that the fourth root of 16 is 2, you then apply the negative sign to get -2.
• Handling Negative Exponents: When dealing with negative signs in algebraic expressions, it's important to handle them carefully. They can change the entire value of your expression, so make sure to apply them correctly after performing the initial calculations.

Negative numbers flip the direction of the number line. For example, -2 is to the left of 0, while +2 is to the right. When you work with equations involving roots, always take note if the expression initially has a negative sign or if the root itself may lead to different interpretations, such as in even roots.

Exponents and Roots

Exponents and roots are fundamental components of algebra. They help simplify expressions and solve equations.

• Exponents Explained: An exponent denotes how many times a number, known as the base, is multiplied by itself. For instance, in the expression \(2^4\), 2 is multiplied by itself four times to yield 16.
• Understanding Roots: A root, like the fourth root \(\sqrt[4]{}\), is the opposite of raising a number to a power. Finding the fourth root of 16 involves determining which number, when raised to the fourth power, results in 16, which is 2 in this case.

Roots enable us to "undo" exponents. In algebra, knowing when to use exponents and their roots, like squares, cubes, or fourths, is crucial. Roots with even indices can have both positive and negative results, but in this specific problem, the negative sign is external and applied after finding the root.

Algebraic Expressions

Algebraic expressions form the backbone of algebra itself. They are combinations of variables, numbers, and operations. Here’s how they play out in this problem:

• Variables and Constants: Although the original exercise uses constants (numbers) rather than variables, the concept is similar. An algebraic expression can often include letters to represent numbers, making them versatile for various problems.
• Operations within Expressions: Algebraic expressions require us to add, subtract, multiply, divide, or use roots and powers, as seen in \(-\sqrt[4]{16}\). Each part works together to solve for the desired values, demonstrating their intricate nature.

Working with algebraic expressions involves fluency with manipulating and simplifying them, understanding their components like coefficients and terms, and applying rules consistently. Mastering these concepts eases the interpretation and solving of more complex equations and expressions.