Problem 110 - Intermediate Algebra

Question

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \sqrt[3]{7} \cdot \sqrt[3]{11}=\sqrt[3]{18} $$

Step-by-Step Solution

Answer

False

Step 1: Understanding Cube Roots

A cube root of a number is a value that, when multiplied by itself twice, gives the original number. That means if we have \( x = \sqrt[3]{y} \), then \( x^3 = y \). Cube roots follow the property that \( \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} \).

Step 2: Apply Cube Root Property

Using the cube root property, \( \sqrt[3]{7} \cdot \sqrt[3]{11} = \sqrt[3]{7 \cdot 11} \). Calculate \( 7 \cdot 11 = 77 \). Therefore, we have \( \sqrt[3]{7} \cdot \sqrt[3]{11} = \sqrt[3]{77} \).

Step 3: Compare the Expressions

Now, compare \( \sqrt[3]{77} \) and \( \sqrt[3]{18} \). These two expressions are not equal because 77 is not equal to 18, meaning their cube roots cannot be equal.

Step 4: Concluding the Truth Value

Since \( \sqrt[3]{7} \cdot \sqrt[3]{11} = \sqrt[3]{77} \) and \( \sqrt[3]{77} eq \sqrt[3]{18} \), the statement \( \sqrt[3]{7} \cdot \sqrt[3]{11} = \sqrt[3]{18} \) is false.

Key Concepts

Properties of Radicals

Radicals have unique properties that help simplify expressions involving roots, such as square roots or cube roots. Understanding these properties can help navigate expressions like cube roots more easily.

• Product Property: One of the main properties is the product property of radicals. It states that \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \). This means if two numbers are under the same root, you can multiply them together under one root.
• Simplification: This property also helps in simplifying radical expressions, making them easier to work with, especially in equations.
• Non-negativity: It’s important to note that these properties are devised under the assumption that all numbers under a radical are non-negative real numbers, ensuring that radical expressions remain within the realm of real numbers.

This exercise uses the product property of radicals to demonstrate the simplification of cube roots, showing how two separate cube roots could combine through multiplication.

Multiplication of Radicals

When multiplying radicals, especially cube roots, the product property allows us to combine them into a single radical. This method simplifies complex-looking expressions.

Example: Cube Roots

To multiply radicals like cubes, simply use their product property. For instance, instead of computing each root separately:

• First, multiply the numbers inside the radicals: \( 7 \cdot 11 = 77 \).
• Apply the combined radical: \( \sqrt[3]{7} \cdot \sqrt[3]{11} = \sqrt[3]{77} \).

This technique helps in handling larger numbers within a single radical, making calculations easier.

Comparing Results

The given statement \( \sqrt[3]{7} \cdot \sqrt[3]{11} = \sqrt[3]{18} \) uses multiplication improperly. By correctly applying the property, we see \( \sqrt[3]{77} eq \sqrt[3]{18} \), indicating the statement is false. This comparison highlights the importance of accurate application of multiplication rules for radicals.

Real Numbers

Understanding real numbers is fundamental when working with radicals, including square and cube roots. Real numbers consist of both rational and irrational numbers and cover everything on a number line.

• Rational Numbers: These are numbers expressed as fractions, like \( \frac{1}{2} \) or whole numbers like 4. Their decimal expansions terminate or repeat.
• Irrational Numbers: These cannot be written as simple fractions. Numbers like \( \pi \) and \( \sqrt{2} \) fall into this category. Their decimal expansion goes on without repeating.

Radicals such as cube roots commonly result in both rational and irrational numbers, depending on the number under the root.

By considering radicals as specific types of real numbers, students can understand how cube roots like \( \sqrt[3]{77} \) and \( \sqrt[3]{18} \) fit into the broader spectrum of real numbers, all while recognizing their separate identities since 77 and 18 themselves are distinct.