Find the simplified product $\sqrt[3]{9 x^4} \cdot \sqrt[3]{3 x^8}$
Answer (2)
The simplified product of 3 9 x 4 ⋅ 3 3 x 8 is 3 x 4 . This is achieved by combining the cube roots and simplifying the expression inside. Finally, taking the cube root of the resulting terms gives us the final answer.
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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 = 3 x 4 .
The simplified product is 3 x 4 .
Explanation
Understanding the Problem We are given the expression 3 9 x 4 ⋅ 3 3 x 8 . Our goal is to simplify this expression into its simplest form.
Combining Cube Roots To simplify the expression, we can use the property that n a ⋅ n b = n a ⋅ b . Applying this property, we have 3 9 x 4 ⋅ 3 3 x 8 = 3 ( 9 x 4 ) ( 3 x 8 )
Simplifying the Expression 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
Taking 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 two irregular objects using cube roots. Simplifying expressions like this helps in combining those volumes to find the total volume more efficiently. This is useful in fields like engineering, where calculating volumes and combining them is a common task. Understanding how to simplify radical expressions makes these calculations easier and more accurate.
