Problem 71 - Algebra and Trigonometry

Question

Simplify the radical expressions in Exercises \(67-74\) if possible. $$ \sqrt[3]{9} \cdot \sqrt[3]{6} $$

Step-by-Step Solution

Answer

\(\sqrt[3]{54}\)

Step 1: Apply Cube Root Rule

According to the cube root rule, we can combine the cube root of 9 and the cube root of 6 into one single cube root of their product: \(\sqrt[3]{9} \cdot \sqrt[3]{6} = \sqrt[3]{9 \cdot 6}\).

Step 2: Calculate The Product

Now calculate the product of 9 and 6: \(9 \cdot 6 = 54\). Replace \(9 \cdot 6\) with 54 in the cube root: \(\sqrt[3]{9 \cdot 6} = \sqrt[3]{54}\).

Step 3: Simplify The Cube Root

Since 54 does not have a perfect cube as a factor, it cannot be simplified further. Therefore, the simplified expression is \(\sqrt[3]{54}\).

Key Concepts

Cube Root Rule

Understanding the cube root rule is essential when working with radical expressions, especially when simplifying them. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because when 2 is multiplied three times (\( 2 \times 2 \times 2 = 8 \)), we obtain 8.

When simplifying expressions involving multiple cube roots, the cube root rule can be applied. This rule allows you to multiply the numbers inside the radicals before finding the cube root of the resulting product. From our exercise example, \(\sqrt[3]{9} \cdot \sqrt[3]{6}\) becomes \(\sqrt[3]{9 \cdot 6}\). This consolidation is possible because of the associative property of multiplication, which applies to cube roots just as it does to regular multiplication.

Radical Simplification

Radical simplification is the process of making a radical expression as