Problem 116 - Intermediate Algebra

Question

Find and correct the error. See the Concept Check in this section. $$ \frac{\sqrt[4]{16}}{\sqrt{4}}=\sqrt[4]{\frac{16}{4}}=\sqrt[4]{4} $$

Step-by-Step Solution

Answer

The error was in incorrectly applying radical division rules.

Step 1: Evaluate the Radicals Separately

First, evaluate both the numerator and the denominator of \( \frac{\sqrt[4]{16}}{\sqrt{4}} \) separately. The fourth root of 16, \( \sqrt[4]{16} \), is equal to 2, because \( 2^4 = 16 \). The square root of 4, \( \sqrt{4} \), is also 2, because \( 2^2 = 4 \).

Step 2: Find the Quotient of Radicals

Now, express the fraction in the form of these evaluated radicals: \( \frac{2}{2} = 1 \). This shows that \( \frac{\sqrt[4]{16}}{\sqrt{4}} = 1 \).

Step 3: Verify Using Radical Property

Re-evaluate the combined radical expression \( \sqrt[4]{\frac{16}{4}} \). Since \( \frac{16}{4} = 4 \), take the fourth root of 4: \( \sqrt[4]{4} \) which evaluates to approximately 1.41, not matching with \( \frac{2}{2} = 1 \).

Step 4: Locate the Error in Calculation

The error in the original expression is in assuming that \( \frac{\sqrt[4]{16}}{\sqrt{4}} = \sqrt[4]{\frac{16}{4}} \). These expressions are not equivalent: the step of combining the radicals was incorrect because the fourth root of the fraction is not equal to the division of the individual roots.

Key Concepts

Error Correction in Radical Expressions

When dealing with radical expressions, it's crucial to ensure proper calculations are made to avoid errors. In the original exercise, the error occurred when the radicals were combined incorrectly.

• Combining radicals with different indices directly, as attempted in the original problem, leads to incorrect evaluations.
• Each radical should be evaluated separately before comparing or performing any operations on them.

The misunderstanding typically arises from neglecting the fundamental differences between the operations of radicals. Properly separating or combining them plays a significant role in accurate problem-solving. By realizing that individual roots must be simplified before attempting to combine them, errors like the one above can be avoided.

Understanding Radical Properties

Radical expressions involve roots such as square roots, cube roots, and fourth roots, each with its own set of properties. These properties guide how radicals can be manipulated to solve equations accurately.

• The product property states that the root of a product is the product of the roots: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)
• The quotient property is similar: the root of a quotient is the quotient of the roots. However, as shown in the error correction, one must be cautious when combining different types of radicals in a quotient.

It's also important to note that these properties do not apply when the radicals have different indices, such as a square root and a fourth root. Recognizing when these properties hold is key to applying them correctly.

Evaluating Radicals Correctly

Evaluating radicals involves simplifying the root expressions accurately. Each radical should be approached based on its specific index.To evaluate radicals like \( \sqrt[4]{16} \) and \( \sqrt{4} \):

• Find the root index: Determine whether you are finding a square root, cube root, fourth root, etc.
• Identify the value: Simplify each number according to its root. For example, the fourth root of 16, or \( 2^4 = 16 \), while \( \sqrt{4} \) simplifies to \( 2^2 = 4 \).

Always perform these evaluations separately before performing any operations between them, such as division or multiplication. This ensures a correct and detailed understanding of each part of the expression.