Problem 50 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(2 \sqrt[3]{2}(3 \sqrt[3]{6}-4 \sqrt[3]{5})\)

Step-by-Step Solution

Answer

The product in simplest radical form is \(12\sqrt[3]{3} - 8\sqrt[3]{10}\).

Step 1: Distribute the expression

Start by applying the distributive property to the expression: \(2 \sqrt[3]{2}(3 \sqrt[3]{6} - 4 \sqrt[3]{5})\).This involves distributing \(2 \sqrt[3]{2}\) to each term inside the parentheses:\[2 \sqrt[3]{2} \times 3 \sqrt[3]{6} - 2 \sqrt[3]{2} \times 4 \sqrt[3]{5}\]

Step 2: Simplify each term

Simplify each product from Step 1 separately:1. Simplify \(2 \sqrt[3]{2} \times 3 \sqrt[3]{6}\): - Multiply the constants and the cube roots: \(2 \times 3 = 6\), - Combine the cube roots: \(\sqrt[3]{2} \times \sqrt[3]{6} = \sqrt[3]{12}\). Simplified first term: \(6 \sqrt[3]{12}\).2. Simplify \(2 \sqrt[3]{2} \times 4 \sqrt[3]{5}\): - Multiply the constants and the cube roots: \(2 \times 4 = 8\), - Combine the cube roots: \(\sqrt[3]{2} \times \sqrt[3]{5} = \sqrt[3]{10}\). Simplified second term: \(8 \sqrt[3]{10}\).

Step 3: Write the simplified expression

Combine the terms obtained in Step 2 to write the final expression:\[6 \sqrt[3]{12} - 8 \sqrt[3]{10}\]

Step 4: Simplify the radicals if possible

Determine if the cube roots can be further simplified by factoring out cubes:- \(\sqrt[3]{12}\) can be simplified further as \(\sqrt[3]{2^2 \times 3} = 2 \sqrt[3]{3}\).- \(\sqrt[3]{10}\) remains unchanged as \(\sqrt[3]{10}\) because 10 is already in its simplest form.Thus, the expression becomes:\[6(2\sqrt[3]{3}) - 8 \sqrt[3]{10} = 12\sqrt[3]{3} - 8\sqrt[3]{10}\]

Step 5: Final Expression

Verify that all terms are in their simplest form. The components \(\sqrt[3]{3}\) and \(\sqrt[3]{10}\) are already in simplest radical form, so the final expression is:\[12\sqrt[3]{3} - 8\sqrt[3]{10}\]

Key Concepts

Simplest Radical Form

Algebraic expressions can often be expressed in a more manageable and "simplest" form and this is key to understanding the problem's solution. The simplest radical form means writing a radical expression in a way that no further simplification is possible. Typically, this includes removing any perfect square factors, or in this case, perfect cubes, from under the radical sign.
To simplify the radical, factor the number into primes. If dealing with cube roots, look for groups of three identical factors, as these can be "pulled out" from the radical. For example, in the expression \(\sqrt[3]{12}\), we factor 12 as \(2^2 \times 3\). Since we don't have a group of three, we only simplify by pulling any cube out that's already there or acknowledging when a further simplification isn't possible.

• For \(\sqrt[3]{12}\), we get \(2 \sqrt[3]{3}\).
• \(\sqrt[3]{10}\) is already in its simplest form with no perfect cube factors.

These simplifications show that the final solution is in its simplest radical form through detailed factorization.

Distributive Property

The distributive property is an essential principle in algebra that allows you to simplify complex expressions. It involves multiplying each term inside a parenthesis by another term outside the parenthesis.
In algebra, this property is expressed as: \(a(b + c) = ab + ac\). This means you distribute the term outside the parenthesis to each term inside.
In the given problem, we applied the distributive property to \(2 \sqrt[3]{2}(3 \sqrt[3]{6} - 4 \sqrt[3]{5})\). By distributing \(2 \sqrt[3]{2}\) to both \(3 \sqrt[3]{6}\) and \(-4 \sqrt[3]{5}\), you separate and handle each multiplication independently:

• First with \(3 \sqrt[3]{6}\), giving \(6 \sqrt[3]{12}\)
• Then with \(-4 \sqrt[3]{5}\), yielding \(-8 \sqrt[3]{10}\)

This method simplifies the original problem effectively and illustrates why the distributive property is such a powerful tool in algebraic manipulation.

Cube Roots

Understanding cube roots is crucial when solving radical expressions involving cubes. A cube root of a number is a value that, when used in a multiplication by itself twice, gives that original number. Denoted by \(\sqrt[3]{x}\), if \(x = y^3\), then \(y\) is the cube root of \(x\). So, \(\sqrt[3]{8} = 2\), because \(2^3 = 8\).
In many algebraic expressions, identifying cube roots helps simplify the expression by breaking down products or factoring out cube numbers. With cube roots, you want to check for any cubic factors that can be simplified further. When working with our expression, \(\sqrt[3]{12}\) and \(\sqrt[3]{10}\), we sought further simplification:

• \(\sqrt[3]{12}\) simplified to \(2 \sqrt[3]{3}\) by extracting any potential cubic factors.
• \(\sqrt[3]{10}\) had no simplification as \(10\) lacks cubic factors, meaning it's already in simplest form.

Emphasizing cube roots and their properties equips you with the necessary tools to decode and rearrange radicals, expressing results neatly and in their simplest forms.