What is the following sum?
$2\left(\sqrt[3]{16 x^3 y}\right)+4\left(\sqrt[3]{54 x^6 y^5}\right)$
Answer (2)
To simplify the expression 2 ( 3 16 x 3 y ) + 4 ( 3 54 x 6 y 5 ) , we can calculate each term, resulting in 4 x 3 2 y + 12 x 2 y 3 2 y 2 . The final simplified sum is 4 x 3 2 y + 12 x 2 y 3 2 y 2 .
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Simplify each term by factoring out perfect cubes from the radicals.
Combine the simplified terms by adding coefficients of like terms.
The simplified sum is 8 x 3 x y + 24 x 3 y 2 3 y + 8 x 3 2 y + 48 x 3 y 3 2 y + 24 x 2 y 3 2 y 2 + 18 x 3 y 3 2 y 2 .
The final answer is 8 x 3 x y + 24 x 3 y 2 3 y + 8 x 3 2 y + 48 x 3 y 3 2 y + 24 x 2 y 3 2 y 2 + 18 x 3 y 3 2 y 2 .
Explanation
Problem Analysis We are asked to find the sum of the following expressions: 2 ( 3 16 x 3 y ) + 4 ( 3 54 x 6 y 5 ) 4 x ( 3 2 y ) + 12 x 2 y ( 3 2 y 2 ) 8 x ( 3 x y ) + 12 x 3 y 2 ( 3 8 y ) 18 x 3 y ( 3 2 y 2 ) 48 x 3 y ( 3 2 y ) Our objective is to simplify each term and then combine like terms to find the final sum.
Simplifying Each Term Let's simplify each term individually: Term 1: 2 ( 3 16 x 3 y ) = 2 3 8 ⋅ 2 x 3 y = 2 ⋅ 2 x 3 2 y = 4 x 3 2 y Term 2: 4 ( 3 54 x 6 y 5 ) = 4 3 27 ⋅ 2 x 6 y 3 y 2 = 4 ⋅ 3 x 2 y 3 2 y 2 = 12 x 2 y 3 2 y 2 Term 3: 4 x ( 3 2 y ) remains as is. Term 4: 12 x 2 y ( 3 2 y 2 ) remains as is. Term 5: 8 x ( 3 x y ) remains as is. Term 6: 12 x 3 y 2 ( 3 8 y ) = 12 x 3 y 2 ⋅ 2 3 y = 24 x 3 y 2 3 y Term 7: 18 x 3 y ( 3 2 y 2 ) remains as is. Term 8: 48 x 3 y ( 3 2 y ) remains as is.
Combining Like Terms Now, let's add all the simplified terms: 4 x 3 2 y + 12 x 2 y 3 2 y 2 + 4 x ( 3 2 y ) + 12 x 2 y ( 3 2 y 2 ) + 8 x ( 3 x y ) + 24 x 3 y 2 3 y + 18 x 3 y ( 3 2 y 2 ) + 48 x 3 y ( 3 2 y ) Combine like terms: ( 4 x 3 2 y + 4 x 3 2 y ) + ( 12 x 2 y 3 2 y 2 + 12 x 2 y 3 2 y 2 ) + 8 x ( 3 x y ) + 24 x 3 y 2 3 y + 18 x 3 y ( 3 2 y 2 ) + 48 x 3 y ( 3 2 y ) 8 x 3 2 y + 24 x 2 y 3 2 y 2 + 8 x ( 3 x y ) + 24 x 3 y 2 3 y + 18 x 3 y ( 3 2 y 2 ) + 48 x 3 y ( 3 2 y )
Rearranging Terms Rearrange the terms: 8 x 3 2 y + 48 x 3 y ( 3 2 y ) + 24 x 2 y 3 2 y 2 + 18 x 3 y ( 3 2 y 2 ) + 8 x ( 3 x y ) + 24 x 3 y 2 3 y ( 8 + 48 x 2 y ) x 3 2 y + ( 24 x 2 y + 18 x 3 y ) 3 2 y 2 + 8 x ( 3 x y ) + 24 x 3 y 2 3 y 8 x 3 x y + 24 x 3 y 2 3 y + 8 x 3 2 y + 48 x 3 y 3 2 y + 24 x 2 y 3 2 y 2 + 18 x 3 y 3 2 y 2
Final Sum The final sum is: 8 x 3 x y + 24 x 3 y 2 3 y + 8 x 3 2 y + 48 x 3 y 3 2 y + 24 x 2 y 3 2 y 2 + 18 x 3 y 3 2 y 2
Examples
This type of algebraic simplification is useful in engineering when dealing with complex formulas involving physical quantities. For example, when calculating the total resistance in a circuit with multiple components or determining the overall heat transfer coefficient in a system, simplifying expressions with radicals and combining like terms can make the calculations more manageable and provide clearer insights into the system's behavior. Simplifying such expressions allows engineers to quickly assess the impact of different parameters on the final result, leading to more efficient designs and better problem-solving.
