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 expression simplifies to 12 b 2 ( 3 2 a ) , which corresponds to option B. This involves simplifying each term and combining like terms. The final answer is B.
;
Simplify each term by factoring out perfect cubes or sixth powers where possible.
Combine the terms with the same radical expressions.
Add the coefficients of the like terms.
The final simplified expression is 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 )
Term 3: 6 b 2 ( 3 2 a )
Term 4: 12 b 2 ( 3 2 a )
Term 5: 6 b 2 ( 6 2 a )
Term 6: 12 b 2 ( 6 2 a )
Combining Like Terms Now, let's add all the simplified 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 Result So, the final expression is:
30 b 2 ( 3 2 a ) + 18 b 2 ( 6 2 a )
Examples
Understanding and simplifying radical expressions is crucial in various fields, such as physics and engineering, where complex calculations involving roots and exponents are common. For instance, when analyzing the behavior of waves or calculating the stress on materials, engineers often encounter expressions with radicals. Simplifying these expressions allows for more efficient computation and a clearer understanding of the underlying phenomena. In finance, similar simplifications can aid in modeling investment growth or calculating returns on assets with fractional exponents.
