Find the simplified product: [tex]\sqrt[3]{9 x^4} \cdot \sqrt[3]{3 x^8}[/tex]
Answer (2)
The expression 3 9 x 4 ⋅ 3 3 x 8 simplifies to 3 x 4 using the property of cube roots to combine and simplify the expressions. After combining and simplifying, we find the cube root of 27 x 12 yields the final result. Thus, the answer is 3 x 4 .
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Combine the cube roots using the property n a ⋅ n b = n a ⋅ b : 3 9 x 4 ⋅ 3 3 x 8 = 3 ( 9 x 4 ) ( 3 x 8 ) .
Simplify the expression inside the cube root: ( 9 x 4 ) ( 3 x 8 ) = 27 x 12 .
Take the cube root: 3 27 x 12 = 3 27 ⋅ 3 x 12 .
Evaluate the cube roots to get the final simplified expression: 3 x 4 .
3 x 4
Explanation
Understanding the Problem We are given the expression 3 9 x 4 ⋅ 3 3 x 8 and we want to simplify it.
Combining Cube Roots To simplify the expression, we can use the property n a ⋅ n b = n a ⋅ b . Applying this property, we get: 3 9 x 4 ⋅ 3 3 x 8 = 3 ( 9 x 4 ) ( 3 x 8 )
Simplifying the Expression Inside the Cube Root Now, we simplify the expression inside the cube root: ( 9 x 4 ) ( 3 x 8 ) = 9 ⋅ 3 ⋅ x 4 ⋅ x 8 = 27 x 4 + 8 = 27 x 12 So, we have: 3 27 x 12
Evaluating the Cube Roots Next, we take the cube root of the simplified expression: 3 27 x 12 = 3 27 ⋅ 3 x 12 We know that 3 27 = 3 because 3 3 = 27 . Also, 3 x 12 = x 3 12 = x 4 . Therefore, 3 27 ⋅ 3 x 12 = 3 ⋅ x 4 = 3 x 4
Final Answer Thus, the simplified product is 3 x 4 .
Examples
Imagine you're calculating the volume of a storage container that is shaped like a cube, but the side lengths are expressed in terms of cube roots. Simplifying expressions like 3 9 x 4 ⋅ 3 3 x 8 helps you find the actual side length, and thus the volume, in a more usable form. This is also useful in physics when dealing with quantities that scale with the cube root of a variable, such as in fluid dynamics or thermodynamics. Simplifying such expressions allows for easier calculations and a better understanding of the relationships between different physical quantities.
