Problem 34 - Elementary and 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 \( 6 + \sqrt[3]{54} \).

Step 1: Apply the Distributive Property

Begin by applying the distributive property to the expression. You want to multiply the cube root of 3 with each term inside the parentheses: \( \sqrt[3]{3} \times 2\sqrt[3]{9} + \sqrt[3]{3} \times \sqrt[3]{18} \).

Step 2: Simplify Each Term Separately

Simplify each term individually. First term: \( 2\sqrt[3]{3} \times \sqrt[3]{9} = 2\sqrt[3]{27} \). For the second term: \( \sqrt[3]{3} \times \sqrt[3]{18} = \sqrt[3]{54} \).

Step 3: Evaluate Cube Roots

Now, simplify the cube roots. For the first term, \( \sqrt[3]{27} = 3 \), so it becomes \( 2 \times 3 = 6 \). For the second term, since \( \sqrt[3]{54} = 54^{1/3} \) which does not simplify further to an integer, it remains \( \sqrt[3]{54} \).

Step 4: Prepare the Final Answer

Combine the simplified terms. The expression simplifies to \( 6 + \sqrt[3]{54} \). This is the simplified form as \( \sqrt[3]{54} \) does not break down further using integer cube roots.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions with operations inside parentheses. It essentially states that multiplying a single term by terms inside a bracket is the same as performing the multiplication individually for each term inside the bracket and then adding or subtracting the results together. This property allows for easier manipulation and simplification of algebraic expressions.

For example, with the expression \[ a(b + c) \], applying the distributive property would involve multiplying \(a\) by both \(b\) and \(c\), giving you \(ab + ac\). This helps break down more complex problems into manageable pieces that can be tackled step by step.

In our original exercise, \( \sqrt[3]{3}(2 \sqrt[3]{9} + \sqrt[3]{18}) \), we applied the distributive property to distribute \( \sqrt[3]{3} \) (the cube root of 3) across each term inside the parentheses. The resulting expression was \( \sqrt[3]{3} \times 2\sqrt[3]{9} + \sqrt[3]{3} \times \sqrt[3]{18} \), simplifying the multiplication and making it easier to apply further simplification techniques.

Exploring Cube Roots

Cube roots are an essential part of mathematics, particularly when dealing with powers and roots. The cube root of a number \( n \), denoted as \( \sqrt[3]{n} \), is a value that, when multiplied three times by itself, gives \( n \). Think of the cube root as "undoing" the cubing process.

For instance, the cube root of 8 is \( 2 \) because \( 2 \times 2 \times 2 = 8 \). Cube roots are useful when simplifying expressions, as they allow you to reduce power-based expressions to simpler terms.

In the exercise, we encountered cube roots such as \( \sqrt[3]{3} \), and noticeably, we had to evaluate \( \sqrt[3]{27} \) and \( \sqrt[3]{54} \). \( \sqrt[3]{27} \) simplifies neatly to 3 since \( 3 \times 3 \times 3 = 27 \). However, \( \sqrt[3]{54} \) is more complex, as 54 is not a perfect cube, thereby requiring the expression to remain in its irrational form \( \sqrt[3]{54} \) in the final answer.

Working with Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the foundation for algebra and can be as simple as a single number or as complex as polynomials with multiple terms. Mastering algebraic expressions involves understanding how these elements interact under various operations.

In the initial expression \( \sqrt[3]{3}(2 \sqrt[3]{9} + \sqrt[3]{18}) \), we worked with cube roots and employed the distributive property to simplify the expression. Each part of an algebraic expression must be handled with care, ensuring each operation such as multiplication or addition is correctly executed.

The process involves identifying each component of the expression, applying the correct mathematical rules like the distributive property, and simplifying each term as much as possible. Only through careful manipulation, step by step, can we arrive at a simplified expression like \( 6 + \sqrt[3]{54} \). By understanding the roles of variables, roots, and operations, you can confidently tackle more complex algebraic tasks.