Find the simplified product: $\sqrt[3]{2 x^5} \cdot \sqrt[3]{64 x^9}$
Answer (2)
To simplify 3 2 x 5 ⋅ 3 64 x 9 , we combine the cube roots and simplify the expression inside. After factoring perfect cubes and simplifying, we arrive at the final expression 4 x 4 3 2 x 2 .
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Combine the cube roots: 3 2 x 5 ⋅ 3 64 x 9 = 3 2 x 5 ⋅ 64 x 9 .
Simplify the expression inside the cube root: 2 x 5 ⋅ 64 x 9 = 128 x 14 .
Factor out perfect cubes: 3 128 x 14 = 3 4 3 ⋅ 2 ⋅ ( x 4 ) 3 ⋅ x 2 .
Simplify: 4 x 4 3 2 x 2 .
4 x 4 3 2 x 2
Explanation
Understanding the Problem We are given the expression 3 2 x 5 ⋅ 3 64 x 9 . Our goal is to simplify this expression. We will use properties of radicals to combine and simplify the terms.
Combining Cube Roots First, we use the property n a ⋅ n b = n a ⋅ b to combine the two cube roots: 3 2 x 5 ⋅ 3 64 x 9 = 3 2 x 5 ⋅ 64 x 9 .
Simplifying Inside the Cube Root Next, we simplify the expression inside the cube root: 2 x 5 ⋅ 64 x 9 = 128 x 14 . So we have 3 128 x 14 .
Factoring Perfect Cubes Now, we factor out perfect cubes from inside the cube root. We know that 128 = 64 ⋅ 2 = 4 3 ⋅ 2 and x 14 = x 12 ⋅ x 2 = ( x 4 ) 3 ⋅ x 2 . Therefore, we can rewrite the expression as: 3 128 x 14 = 3 4 3 ⋅ 2 ⋅ ( x 4 ) 3 ⋅ x 2 .
Separating Perfect Cubes Using the property n a ⋅ b = n a ⋅ n b , we separate the perfect cubes: 3 4 3 ⋅ 2 ⋅ ( x 4 ) 3 ⋅ x 2 = 3 4 3 ⋅ 3 ( x 4 ) 3 ⋅ 3 2 x 2 .
Simplifying Cube Roots Finally, we simplify the cube roots of the perfect cubes: 3 4 3 ⋅ 3 ( x 4 ) 3 ⋅ 3 2 x 2 = 4 x 4 3 2 x 2 .
Final Answer Thus, the simplified expression is 4 x 4 3 2 x 2 .
Examples
Imagine you're designing a storage container, and its volume is determined by the expression 3 2 x 5 ⋅ 3 64 x 9 . Simplifying this expression to 4 x 4 3 2 x 2 helps you understand the relationship between the variable x and the container's volume more clearly. This simplification allows for easier calculations and better optimization of the container's dimensions, ensuring efficient use of space and resources. Understanding radical expressions and their simplification is crucial in various engineering and design applications.
