Problem 52 Simplify the products. Give exac... [FREE SOLUTION]

Chapter 7: Problem 52

Simplify the products. Give exact answers. $$\sqrt[3]{2}(\sqrt[3]{12 x}-\sqrt[3]{2 x})$$

Short Answer

Expert verified

$ \sqrt[3]{24x} - \sqrt[3]{4x} $

Step by step solution

- Distribute $ \sqrt[3]{2} $

Distribute $ \sqrt[3]{2} $ to both terms inside the parentheses: $ \sqrt[3]{2} \times (\sqrt[3]{12x} - \sqrt[3]{2x}) = \sqrt[3]{2} \times \sqrt[3]{12x} - \sqrt[3]{2} \times \sqrt[3]{2x} $

- Apply the property of cube roots

Apply the property $ \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{ab} $ to both terms: $ \sqrt[3]{2} \times \sqrt[3]{12x} = \sqrt[3]{24x} $ and $ \sqrt[3]{2} \times \sqrt[3]{2x} = \sqrt[3]{4x} $

- Simplify the expression

Combine the results from Step 2: $ \sqrt[3]{24x} - \sqrt[3]{4x} $

- Write the final expression

The simplified form of $ \sqrt[3]{2}(\sqrt[3]{12x}-\sqrt[3]{2x}) $ is $ \sqrt[3]{24x} - \sqrt[3]{4x} $

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property

To solve the exercise, we start by applying the distributive property. This property states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. In this example, we distribute the cube root of 2, $ \sqrt[3]{2} $, across the terms inside the parentheses. So, $ \sqrt[3]{2} (\sqrt[3]{12x} - \sqrt[3]{2x}) $ turns into $ \sqrt[3]{2} \times \sqrt[3]{12x} - \sqrt[3]{2} \times \sqrt[3]{2x} $. This step helps break down complex expressions into simpler, manageable parts.

properties of cube roots

Next, understanding the properties of cube roots allows us to simplify further. One key property is that the product of cube roots can be combined: $ \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{ab} $. Applying this to our terms:

$ \sqrt[3]{2} \times \sqrt[3]{12x} = \sqrt[3]{24x} $
$ \sqrt[3]{2} \times \sqrt[3]{2x} = \sqrt[3]{4x} $

Recognizing and using these properties are fundamental to simplifying cube root expressions.

simplifying expressions

The final step involves simplifying our expression to its most reduced form. After distributing and applying the properties of cube roots, combine the results: $ \sqrt[3]{24x} - \sqrt[3]{4x} $. This is the simplest form of the original expression $ \sqrt[3]{2} (\sqrt[3]{12x} - \sqrt[3]{2x}) $. Remember, simplifying expressions often involves multiple steps, but each makes the problem more approachable. Practice these steps to improve proficiency in simplifying complex algebraic expressions involving cube roots.

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91影视

Simplify the products. Give exact answers. $$\sqrt[3]{2}(\sqrt[3]{12 x}-\sqrt[3]{2 x})$$

Short Answer

Step by step solution

- Distribute \( \sqrt[3]{2} \)

- Apply the property of cube roots

- Simplify the expression

- Write the final expression

Key Concepts

distributive property

properties of cube roots

simplifying expressions

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