What is the following sum?
[tex]$3 b^2(\sqrt[3]{54 a})+3(\sqrt[3]{2 a b^6})$[/tex]
A. [tex]$6 b^2(\sqrt[3]{2 a})$[/tex]
B. [tex]$12 b^2(\sqrt[3]{2 a})$[/tex]
C. [tex]$6 b^2(\sqrt[6]{2 a})$[/tex]
D. [tex]$12 b^2(\sqrt[6]{2 a})$[/tex]
Answer (2)
The simplified sum of the expression is 12 b 2 3 2 a , which corresponds to option B. The first term simplifies to 9 b 2 3 2 a and the second term to 3 b 2 3 2 a . Adding both gives 12 b 2 3 2 a .
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Simplify the first term: 3 b 2 ( 3 54 a ) = 9 b 2 ( 3 2 a ) .
Simplify the second term: 3 ( 3 2 a b 6 ) = 3 b 2 ( 3 2 a ) .
Add all the terms: 9 b 2 ( 3 2 a ) + 3 b 2 ( 3 2 a ) + 6 b 2 ( 3 2 a ) + 12 b 2 ( 3 2 a ) + 6 b 2 ( 6 2 a ) + 12 b 2 ( 6 2 a ) .
Combine like terms to get the final answer: 30 b 2 ( 3 2 a ) + 18 b 2 ( 6 2 a ) .
Explanation
Problem Analysis We are asked to find the sum of the following expressions:
3 b 2 ( 3 54 a ) + 3 ( 3 2 a b 6 ) 6 b 2 ( 3 2 a ) 12 b 2 ( 3 2 a ) 6 b 2 ( 6 2 a ) 12 b 2 ( 6 2 a )
Simplifying the terms Let's simplify each term individually.
Term 1: 3 b 2 ( 3 54 a ) = 3 b 2 ( 3 27 ⋅ 2 a ) = 3 b 2 ( 3 3 2 a ) = 9 b 2 ( 3 2 a )
Term 2: 3 ( 3 2 a b 6 ) = 3 ( 3 b 6 3 2 a ) = 3 b 2 ( 3 2 a )
Adding the terms Now, let's add all the terms together:
9 b 2 ( 3 2 a ) + 3 b 2 ( 3 2 a ) + 6 b 2 ( 3 2 a ) + 12 b 2 ( 3 2 a ) + 6 b 2 ( 6 2 a ) + 12 b 2 ( 6 2 a )
Combine like terms:
( 9 + 3 + 6 + 12 ) b 2 ( 3 2 a ) + ( 6 + 12 ) b 2 ( 6 2 a )
30 b 2 ( 3 2 a ) + 18 b 2 ( 6 2 a )
Final Answer Therefore, the sum of the given expressions is 30 b 2 ( 3 2 a ) + 18 b 2 ( 6 2 a ) .
Examples
This type of algebraic simplification can be useful in various fields, such as physics or engineering, where complex formulas need to be simplified for easier calculation or analysis. For instance, when dealing with volumes or areas involving cube roots and square roots, simplifying expressions like this can make the calculations more manageable. Imagine you're calculating the total material needed for a construction project, and you have several components with dimensions involving cube roots and sixth roots. Simplifying the expression first can save time and reduce the chance of errors.
