Problem 22 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[4]{81}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[4]{81}\) is 3.

Step 1: Identify the expression

The expression we need to simplify is \(\sqrt[4]{81}\). This is a fourth root expression.

Step 2: Express the number as a power

Recognize that 81 can be expressed as a power of 3: \(81 = 3^4\). This helps in simplifying the radical expression.

Step 3: Simplify the radical using the power

Since \(\sqrt[4]{81}\) is the fourth root of \(3^4\), this simplifies to \(3\). The fourth root of \(3^4\) is \(3\), because raising \(3\) to the fourth power gives \(81\).

Key Concepts

Fourth Roots

Fourth roots are similar to square roots but involve four factors instead of two. They are denoted by the radical expression \( \sqrt[4]{a} \). This symbol means that we are looking for a number which, when multiplied by itself four times, equals \( a \). For example, finding \( \sqrt[4]{81} \) means determining what number, when used as a factor four times, produces 81. In this particular case, it is 3 because \( 3 \times 3 \times 3 \times 3 = 81 \). Understanding fourth roots is crucial when simplifying expressions in algebra, as they allow us to reduce large numbers or terms to their original, smaller contributors.

Exponents

Exponents provide a way to express repeated multiplication compactly. The notation \( a^n \) means that the number \( a \) is multiplied by itself \( n \) times. In the exercise, understanding \( 81 \) as \( 3^4 \) is key to simplifying the radical expression. Here, the exponent is 4, indicating that the base, 3, appears four times in a multiplication sequence. By expressing a number as a power, it's easier to take roots, as we can directly use the relationship between roots and exponents to simplify the expression. Recognizing this allows us to replace complex radicals with simpler numbers. This simplifies calculations and assists in solving algebraic problems efficiently.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest forms. This typically involves removing radicals where possible, combining like terms, and making other adjustments to present the expression as simply as possible. The exercise of simplifying \( \sqrt[4]{81} \) illustrates this principle well. By recognizing that 81 is \( 3^4 \), we easily find that the expression simplifies to 3. By simplifying expressions, equations become easier to work with and understand. Simplification is a powerful tool in algebra for solving equations, evaluating expressions, and understanding relationships between variables.