Problem 34 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt[3]{3}(2 \sqrt[3]{9}+\sqrt[3]{18}) $$

Step-by-Step Solution

Answer

The simplified expression is \( 6 + 3\sqrt[3]{2} \).

Step 1: Distribute the Cube Root

Begin by distributing the cube root of 3, \( \sqrt[3]{3} \), into the expression inside the parentheses: \( 2\sqrt[3]{9} + \sqrt[3]{18} \). This means you perform the multiplication as follows: \( \sqrt[3]{3} \times 2\sqrt[3]{9} + \sqrt[3]{3} \times \sqrt[3]{18} \).

Step 2: Multiply the Cube Roots

Focus on each part separately: 1. For \( \sqrt[3]{3} \times 2\sqrt[3]{9} \), you multiply the coefficients and the radicands: - Coefficient: \( 2 \times 1 = 2 \) - Radicands: \( \sqrt[3]{3} \times \sqrt[3]{9} = \sqrt[3]{3 \times 9} = \sqrt[3]{27} \). - So, \( \sqrt[3]{3} \times 2\sqrt[3]{9} = 2\sqrt[3]{27} \).2. For \( \sqrt[3]{3} \times \sqrt[3]{18} \), you have: - Radicands: \( \sqrt[3]{3} \times \sqrt[3]{18} = \sqrt[3]{3 \times 18} = \sqrt[3]{54} \). - So, \( \sqrt[3]{3} \times \sqrt[3]{18} = \sqrt[3]{54} \).

Step 3: Simplify the Cube Roots

Simplify each cube root, if possible.1. \( \sqrt[3]{27} = 3 \) because \( 27 = 3^3 \).2. \( \sqrt[3]{54} \) can be simplified by finding prime factors: - \( 54 = 2 \times 3^3 \) - \( \sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = 3\sqrt[3]{2} \).So, the expression becomes \( 2 \times 3 \) \( + 3\sqrt[3]{2} \) which simplifies further to \( 6 + 3\sqrt[3]{2} \).

Step 4: Final Expression

The simplified expression of the entire problem is obtained by combining the results from previous steps: \( 6 + 3\sqrt[3]{2} \). This is the final simplified form of the original expression.

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that helps you simplify expressions involving multiplication over addition or subtraction. Think of it as distributing or handing out a factor to terms within parentheses. This property is especially useful when dealing with complex mathematical expressions, such as those involving radicals or roots.

In our exercise, the distributive property is applied to the expression \( \sqrt[3]{3}(2 \sqrt[3]{9}+\sqrt[3]{18}) \). Here, the cube root \( \sqrt[3]{3} \) is distributed to each term inside the parentheses:

• \( \sqrt[3]{3} \times 2 \sqrt[3]{9} \)
• \( \sqrt[3]{3} \times \sqrt[3]{18} \)

This step helps break down the problem into smaller and more manageable parts. Remember, the distributive property is not limited to numbers. It can be applied to algebraic expressions, including those with roots like cube roots.

Simplifying Radicals

Simplifying radicals involves expressing radical terms like square roots or cube roots in their simplest form. Simplification often requires finding a factor of the radicand (the number inside the root) that is a perfect square or cube, depending on the root.

For example, in our exercise, we simplify the expression \( \sqrt[3]{27} = 3 \) because 27 is a perfect cube: \( 3^3 \). These kinds of simplifications make calculations easier and expressions clearer. When the radicand is not a perfect power, like \( \sqrt[3]{54} \), it can still be simplified to a form such as \( 3\sqrt[3]{2} \) by factoring \( 54 \) into \( 2 \times 3^3 \).

• Find factors of the radicand that can be removed or simplified.
• Use prime factorization as a helpful tool to identify perfect powers.

This process allows easier manipulation of radical expressions and eventual solutions.

Multiplication of Radicals

When multiplying radicals, it is essential to follow specific rules to ensure accurate results. A radical is essentially a root, and when you multiply them, you're typically multiplying their radicands, which are the numbers or expressions inside the radical symbol.

In our example, multiplication of cube roots follows these guidelines. Consider \( \sqrt[3]{3} \times \sqrt[3]{9} \). Here, the radicands 3 and 9 are multiplied to give \( \sqrt[3]{3 \times 9} = \sqrt[3]{27} \). The product of the radicands is then simplified further when possible, like recognizing \( \sqrt[3]{27} \) as 3, its cube root.

• Multiply the coefficients separately from the radicands.
• Ensure the product of radicands is simplified, keeping an eye out for perfect cubes or squares.

Multiplying radicals can seem tricky at first, but by sticking to these basic rules, you ensure your calculations remain clear and correct.