Problem 121 - Elementary and Intermediate Algebra

Question

Explain the mistake in the student's solution shown below. Simplify: \(\sqrt[3]{54}\) $$ \begin{aligned} \sqrt[3]{54} &=\sqrt[3]{27+27} \\ &=\sqrt[3]{27}+\sqrt[3]{27} \\ &=3+3 \\ &=6 \end{aligned} $$

Step-by-Step Solution

Answer

The correct simplification is \\(3\sqrt[3]{2}\\). The error was in trying to split the cube root sum as separate cube roots.

Step 1: Identify the Problem

The mistake lies in incorrectly applying the cube root property. The student applied the rule for addition under cube roots, which is incorrect. The cube root of a sum is not equal to the sum of the cube roots.

Step 2: Correct Understanding of Cube Roots

To simplify \(\sqrt[3]{54}\), find a factorization of 54 that includes a cube number. \(54 = 2 imes 27\), and 27 is a perfect cube (as \(3^3 = 27\)).

Step 3: Separate the Expression

Recognize that \(27 \ imes 2\) can simplify the root: \(\sqrt[3]{27 imes 2}\) = \(\sqrt[3]{27} imes \sqrt[3]{2}\).

Step 4: Calculate the Perfect Cube Root

Knowing \(\sqrt[3]{27} = 3\), simplify this part of the expression. The cube root of 27 is simply 3.

Step 5: Put It All Together

The expression simplifies to \(3 \times \sqrt[3]{2}\). This is the simplest form of the cube root of 54 in terms of elementary cube roots.

Step 6: Conclusion

The correct simplification is \(3\sqrt[3]{2}\). Therefore, the error was in incorrectly applying the cube root property to a sum of numbers.

Key Concepts

Simplifying Cube Roots

When you simplify cube roots, it's about expressing the number neatly in a combination of simpler parts.
For instance, let's consider the cube root of 54. The goal is to break it down into factors, especially looking for any perfect cubes.
54 can be factored as \(2 \times 27\). Here, 27 is a special number known as a perfect cube because it can be expressed as \(3^3\).

• Start by rewriting 54 in terms of these factors: \(\sqrt[3]{54} = \sqrt[3]{27 \times 2}\).
• Due to the property of cube roots, \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\), we can separate them: \(\sqrt[3]{54} = \sqrt[3]{27} \times \sqrt[3]{2}\).
• Since \(\sqrt[3]{27}\) is exactly 3, our expression simplifies to \(3 \times \sqrt[3]{2}\).

This is the tidiest and most straightforward form of a cube root, focusing on simplifying components rather than the original sum.
It's important to always search for those perfect cube factors to help break it down.

Incorrect Cube Root Application

Applying the rules of cube roots incorrectly can lead to mistakes, as shown in the original exercise.
The student incorrectly tried to apply an addition rule with cube roots, assuming \(\sqrt[3]{a+b}\) equals \(\sqrt[3]{a} + \sqrt[3]{b}\).
Unfortunately, this rule doesn't apply to cube roots, nor any roots. The correct way focuses on multiplication:

• The cube root of a product, \(\sqrt[3]{a \times b}\), breaks down into the product of the cube roots: \(\sqrt[3]{a} \times \sqrt[3]{b}\).
• Roots deal more straightforwardly with multiplication or division rather than addition or subtraction.

Understanding these distinctions is vital.
Misapplying root properties can lead to answers that don't reflect the true simplification of the mathematical expression you'll find in this lesson.

Cube Root Properties

Cube roots have unique properties that differentiate them from other root types.
Comprehending these properties is essential in correctly simplifying expressions containing cube roots:

• If you have a product in the form of \(\sqrt[3]{a \times b}\), it simplifies into \(\sqrt[3]{a} \times \sqrt[3]{b}\). This is a core property of cube roots and applies to all positive numbers \(a\) and \(b\).
• Another property is the simplification of perfect cubes. Any perfect cube, say \(n^3\), will simplify to \(n\), because the cube and cube root are inverse operations.
• Cube roots only work naturally with multiplication and division, not with addition or subtraction.

By mastering these properties, you'll navigate cube roots effortlessly.
This understanding will not only correct errors but also build a foundation for more advanced mathematical concepts. Cube roots might seem tricky, but once you grasp their basic properties, they become much easier to handle!