Combine and simplify the following radical expression: [tex]$\sqrt[3]{6} \cdot \sqrt[3]{4}$[/tex]
Answer (2)
The expression 3 6 ⋅ 3 4 simplifies to 2 3 3 by first combining the radicals and then simplifying the radical expression based on prime factorization. The process involves multiplying the terms inside the radicals and identifying perfect cubes. Finally, we isolate and simplify each component of the radical expression.
;
Use the property n a ⋅ n b = n ab to combine the radicals: 3 6 ⋅ 3 4 = 3 6 × 4 .
Multiply the numbers inside the radical: 6 × 4 = 24 , so we have 3 24 .
Factor 24 to find perfect cubes: 24 = 2 3 × 3 .
Simplify the radical expression: 3 2 3 × 3 = 3 2 3 ⋅ 3 3 = 2 3 3 .
The final simplified expression is 2 3 3 .
Explanation
Understanding the Problem We are given the expression 3 6 ⋅ 3 4 . Our goal is to combine and simplify this expression into its simplest radical form.
Combining the Radicals To simplify the expression, we can use the property that n a ⋅ n b = n ab . Applying this property, we get: 3 6 ⋅ 3 4 = 3 6 × 4
Multiplying the Numbers Now, we multiply the numbers inside the radical: 6 × 4 = 24 So, we have 3 24 .
Prime Factorization Next, we want to simplify 3 24 by finding the prime factorization of 24. We have: 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2 3 × 3 So, 24 = 2 3 × 3 .
Separating the Radical Now we can rewrite the radical as: 3 24 = 3 2 3 × 3 Using the property n ab = n a ⋅ n b , we can separate the radical: 3 2 3 × 3 = 3 2 3 ⋅ 3 3
Simplifying the Perfect Cube Since 3 2 3 = 2 , we have: 3 2 3 ⋅ 3 3 = 2 ⋅ 3 3 Thus, the simplified expression is 2 3 3 .
Final Answer Therefore, the simplified form of the given expression is 2 3 3 .
Examples
Radical expressions are useful in various fields such as engineering, physics, and computer graphics, especially when dealing with distances, areas, and volumes. For instance, when calculating the length of the diagonal of a cube with side length s , the diagonal is s 3 . Simplifying radical expressions makes these calculations easier and more intuitive. In computer graphics, radical expressions are used in calculating lighting and shading effects, ensuring that objects appear realistic.
