Problem 14 - Intermediate Algebra

Question

For Problems \(1-14\), multiply and simplify where possible. $$ (9 \sqrt[3]{6})(2 \sqrt[3]{9}) $$

Step-by-Step Solution

Answer

The simplified result is \(54\sqrt[3]{2}\).

Step 1: Identify the Multiplication

The expression given is \((9 \sqrt[3]{6})(2 \sqrt[3]{9})\). We need to multiply two coefficients and two cube roots.

Step 2: Multiply the Coefficients

Multiply the coefficients: \(9\) and \(2\). This gives us \(18\).

Step 3: Multiply the Cube Roots

Multiply the cube roots: \(\sqrt[3]{6} \times \sqrt[3]{9}\). According to the property of cube roots, \(\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{ab}\). Apply this to get \(\sqrt[3]{54}\).

Step 4: Simplify the Cube Root

Simplify \(\sqrt[3]{54}\) if possible. Note that \(54 = 27 \times 2\) and \(27 = 3^3\), so we have \(\sqrt[3]{3^3 \times 2}\). The cube root of \(3^3\) simplifies to \(3\), so we get \(3\sqrt[3]{2}\).

Step 5: Combine the Results

Combine the results from the coefficients and cube root simplification: \(18 \times 3\sqrt[3]{2}\) which is equal to \(54\sqrt[3]{2}\).

Key Concepts

Cube Roots

Understanding cube roots is foundational when simplifying expressions involving exponents. A cube root asks the question: what number, when multiplied by itself three times, gives us the original number? For example, the cube root of 27 is 3 because when we multiply 3 by itself three times (i.e., \(3 \times 3 \times 3\)), we get 27.
When you see an expression like \(\sqrt[3]{A}\), it signifies the cube root of A. In our exercise, we dealt with cube roots like \(\sqrt[3]{6}\) and \(\sqrt[3]{9}\). These can be multiplied together under certain rules, which leads us to the next concept of multiplication of radicals.

Simplification

Simplification allows us to reduce expressions to their most basic form, making them easier to work with. When simplifying cube roots, it's essential to identify perfect cubes that can be factored out.
Take \(\sqrt[3]{54}\). It can be expressed as \(\sqrt[3]{27 \times 2}\); since 27 is a perfect cube \((3^3)\), you simplify \(\sqrt[3]{27}\) to 3. Hence, our expression becomes \(3\sqrt[3]{2}\).
This simplification process helps reduce the expression's complexity and makes calculations more manageable. Simplification not only refines expressions but often provides insights, especially in higher mathematics.

Multiplication of Radicals

Multiplying cube roots follows a straightforward principle. Recall that the multiplication rule of cube roots states that \(\sqrt[3]{A} \times \sqrt[3]{B} = \sqrt[3]{AB}\). This means you can combine the numbers inside the cube roots and then find the cube root of the product.
For our problem, \(\sqrt[3]{6} \times \sqrt[3]{9}\) became \(\sqrt[3]{54}\), by multiplying the numbers inside the cube roots. This step is crucial in many algebraic processes: it streamlines expressions and prepares them for simplification.
In summary, understanding the multiplication of radicals involves recognizing that we must look within the radicals, multiply the contents, and then find a unified cube root, just as we did in the exercise.