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)
Simplify each cube root term by factoring out perfect cubes.
Combine the simplified terms.
Group like terms together.
The final simplified expression is ( 8 x + 48 x 3 y ) 3 2 y + ( 24 x 2 y + 16 x 3 y ) 3 2 y 2 + 8 x ( 3 x y ) + 12 x 3 y 2 ( 3 6 y ) .
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 6 y ) 16 x 3 y ( 3 2 y 2 ) 48 x 3 y ( 3 2 y )
Our goal is to simplify each term and then combine like terms to find the 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 ) - already simplified
Term 4: 12 x 2 y ( 3 2 y 2 ) - already simplified
Term 5: 8 x ( 3 x y ) - already simplified
Term 6: 12 x 3 y 2 ( 3 6 y ) - already simplified
Term 7: 16 x 3 y ( 3 2 y 2 ) - already simplified
Term 8: 48 x 3 y ( 3 2 y ) - already simplified
Adding the Terms Now, let's add all the simplified terms together:
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 ) + 12 x 3 y 2 ( 3 6 y ) + 16 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 ) + 12 x 3 y 2 ( 3 6 y ) + 16 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 ) + 12 x 3 y 2 ( 3 6 y ) + 16 x 3 y ( 3 2 y 2 ) + 48 x 3 y ( 3 2 y )
Combining Like Terms Rearrange the terms to group similar expressions:
8 x 3 2 y + 48 x 3 y ( 3 2 y ) + 24 x 2 y 3 2 y 2 + 16 x 3 y ( 3 2 y 2 ) + 8 x ( 3 x y ) + 12 x 3 y 2 ( 3 6 y )
Combine the terms with 3 2 y and 3 2 y 2 :
( 8 x + 48 x 3 y ) 3 2 y + ( 24 x 2 y + 16 x 3 y ) 3 2 y 2 + 8 x ( 3 x y ) + 12 x 3 y 2 ( 3 6 y )
Final Result So, the final simplified expression is:
( 8 x + 48 x 3 y ) 3 2 y + ( 24 x 2 y + 16 x 3 y ) 3 2 y 2 + 8 x ( 3 x y ) + 12 x 3 y 2 ( 3 6 y )
Examples
This type of simplification can be used in engineering when dealing with volumes or scaling factors that involve cube roots. For example, if you are designing a container and need to calculate the total volume of several components, each with a volume expressed as a cube root, you would use these simplification techniques to combine the volumes and optimize the design. Similarly, in physics, when dealing with quantities that scale with the cube root of a variable, such as the radius of a sphere given its volume, these techniques can help in simplifying complex expressions and making calculations more manageable. Understanding how to simplify and combine these expressions allows for more efficient and accurate problem-solving in these fields.
The sum simplifies to 4 x 3 2 y + 12 x 2 y 3 2 y 2 . Each term was simplified using the properties of cube roots before combining them. The final expression reflects combined terms that cannot be further simplified due to different radicands.
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