Problem 87 - Intermediate Algebra

Question

Evaluate each radical expression, if possible, without using a calculator. See Example 8. $$ \sqrt[4]{81} $$

Step-by-Step Solution

Answer

The fourth root of 81 is 3.

Step 1: Understand the Expression

We are tasked with evaluating the expression \( \sqrt[4]{81} \). This is a fourth root, meaning we need to find a number that, when raised to the power of 4, equals 81.

Step 2: Express 81 as a Power

First, identify whether 81 can be expressed as a power of a smaller integer. We know that \( 81 = 3^4 \) because \( 3 \times 3 = 9 \), \( 9 \times 3 = 27 \), and \( 27 \times 3 = 81 \).

Step 3: Take the Fourth Root of Both Sides

Next, apply the fourth root to the expression: \( \sqrt[4]{81} = \sqrt[4]{3^4} \).

Step 4: Simplify the Fourth Root

The fourth root of \( 3^4 \) is simply 3 because raising 3 to the fourth power gives back the original number, 81.

Key Concepts

Fourth Roots

The concept of a fourth root can be thought of as the opposite of raising a number to the fourth power. When we look at the expression \( \sqrt[4]{81} \), we need to determine what number raised to the power of 4 will result in 81.

• Fourth roots are a special kind of radical expression focusing on powers of four.
• To find \( \sqrt[4]{81} \), we solve for a number \( x \) that satisfies \( x^4 = 81 \).

An effective strategy to find this number is by expressing 81 as a power. Writing 81 as \( 3^4 \) helps to see that the fourth root of \( 3^4 \) is simply 3. Therefore, \( \sqrt[4]{81} = 3 \).
Breaking down a number into its roots involves understanding its factors and acknowledging how these factors can be grouped into powers of four.

Powers and Exponents

Powers and exponents are fundamental concepts in mathematics that describe repeated multiplication. In the case of evaluating \( \sqrt[4]{81} \), they play a crucial role.

• A power indicates how many times a number is multiplied by itself.
• An exponent is a small number placed to the upper right of a number, indicating the power.

The expression \( 3^4 \) means 3 multiplied by itself four times: \( 3 \times 3 \times 3 \times 3 \).
To break it down: - \( 3 \times 3 = 9 \)- then \( 9 \times 3 = 27 \)- and \( 27 \times 3 = 81 \), giving \( 3^4 = 81 \).
Understanding how to simplify and manipulate powers is critical in working with expressions, especially when performing operations like finding roots.

Simplifying Expressions

Simplifying expressions is about rewriting them in a simpler or more convenient form without changing their value. For \( \sqrt[4]{81} \), simplifying involves expressing the number inside the radical in an easier form to find the root.

• It involves recognizing patterns or structures such as expressions that can be represented as powers of numbers.
• In the exercise, it meant rewriting 81 as \( 3^4 \) to easily identify its fourth root.

By understanding the properties of powers and how to deconstruct numbers into more manageable parts, we can simplify complex expressions.
The main goal is to break down these expressions step-by-step, making them more digestible and leading naturally to the solution. This encourages a deeper understanding of the process rather than just finding "the answer."