Problem 88 - Elementary and Intermediate Algebra

Question

Evaluate each radical expression, if possible, without using a calculator. See Example 8 . $$ \sqrt[6]{64} $$

Step-by-Step Solution

Answer

The sixth root of 64 is 2.

Step 1: Identify the Radical Expression

The expression given is \( \sqrt[6]{64} \). This is a sixth root, meaning we need to find a number which, when raised to the power of 6, equals 64.

Step 2: Recognize the Form of the Number

64 is a number that can be expressed as a power of a smaller base number. Recognizing this will help simplify the radical expression.

Step 3: Express the Radicand as a Power

Note that \( 64 = 2^6 \). This means the expression inside the radical, 64, can be rewritten as \( 2^6 \).

Step 4: Apply Radical Exponent Rules

To evaluate \( \sqrt[6]{2^6} \), we apply the rule that \( \sqrt[n]{a^n} = a \). Here \( n = 6 \) and \( a = 2 \), thus \( \sqrt[6]{2^6} = 2 \).

Step 5: Confirm the Solution

Verify the result by calculating \( 2^6 \). Since \( 2^6 = 64 \), the solution \( 2 \) is correct.

Key Concepts

Radical Expression

A radical expression involves a root symbol, typically recognized as a square root, cube root, fourth root, or higher roots. These symbols are used to denote numbers that, when raised to a specific power, return the number under the root. In this context, considering the expression \( \sqrt[6]{64} \), the root is called a sixth root.
To comprehend a radical expression, break it down into parts:

• The index: This is the small number written just outside and to the left of the radical sign, indicating the degree of the root. In \( \sqrt[6]{64} \), 6 is the index.
• The radicand: This is the number or expression inside the radical symbol. In our example, 64 is the radicand.

These components help us understand how the expression works and how to simplify it. The goal with any radical expression is often to find the value that, when multiplied by itself as many times as the index indicates, results in the radicand. For instance, what number, raised to the sixth power gives us 64? This is precisely what the radical expression is asking.

Power of a Number

The power of a number is a way to express repeated multiplication of a number by itself. It is represented as \( a^n \), where \( a \) is the base, and \( n \) is the exponent, showing the number of times the base is used as a factor.
Consider the example \( 64 = 2^6 \):

• The base \( (2) \) is multiplied by itself 6 times.
• Thus, \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\).

Understanding how numbers can be expressed in this form helps simplify problems involving roots and radicals. With the exponents, recognizing the power of a number allows you to simplify roots such as in \( \sqrt[6]{64}\). By expressing 64 as \(2^6\), it becomes apparent how the radical expression simplifies. This linkage between powers and roots is fundamental in checking and verifying our mathematical calculations and simplifying expressions efficiently. The equation emphasizes the relationship between the radical form and exponential form, permitting deeper insight into algebraic structures.

Radical Exponent Rules

Radical exponent rules offer a systematic approach for simplifying expressions involving roots and powers. These rules reveal the symbiotic relationship between radicals and exponents, which can simplify intricate calculations.
When considering a radical like \( \sqrt[6]{64} \), expressing the radicand as a power (like \( 2^6 \)) allows us to apply rules that simplify the operation significantly. The key rule at work here is:

• \( \sqrt[n]{a^n} = a \), where \( n \) is the index of the root. This rule simply states that if the power inside the root matches the root's index, the result is the base \( a \).

For instance, applying \( \sqrt[6]{2^6} = 2 \), succinctly illustrates this principle. It shows how exponents and their corresponding roots effectively "cancel" each other, reverting to the base, simplifying otherwise complex expressions in equations.
This rule is crucial for solving exponential and radical expressions. Grasping how these exponent rules apply provides powerful tools to tackle progressively challenging algebra problems with simplicity and clarity. Understanding these principles not only aids in algebra but lays a foundation for more advanced mathematical studies.