What is the following sum?
[tex]$5 x\left(\sqrt[3]{x^2 y}\right)+2\left(\sqrt[3]{x^5 y}\right)$[/tex]
A. [tex]$7 x\left(\sqrt[6]{x^2 y}\right)$[/tex]
B. [tex]$7 x^2\left(\sqrt[6]{x y^2}\right)$[/tex]
C. [tex]$7 x^2\left(\sqrt[3]{x y^2}\right)$[/tex]
D. [tex]$7 x\left(\sqrt[3]{x^2 y}\right)$[/tex]
Answer (2)
Rewrite each term using fractional exponents.
Combine the like terms.
Simplify the expression by adding the coefficients of the like terms.
The final answer is 14 x 3 x 2 y + 7 x 6 x 2 y + 7 x 2 6 x y 2 + 7 x 2 3 x y 2 .
Explanation
Rewrite the terms with fractional exponents We are asked to find the sum of the following terms:
5 x ( 3 x 2 y ) + 2 ( 3 x 5 y ) 7 x ( 6 x 2 y ) 7 x 2 ( 6 x y 2 ) 7 x 2 ( 3 x y 2 ) 7 x ( 3 x 2 y )
Our goal is to simplify this expression by combining like terms. To do this, we will first rewrite each term using fractional exponents.
Express terms with fractional exponents Let's rewrite each term using fractional exponents:
5 x ( 3 x 2 y ) = 5 x ( x 2 y ) 3 1 = 5 x ( x 3 2 y 3 1 ) = 5 x 3 5 y 3 1
2 ( 3 x 5 y ) = 2 ( x 5 y ) 3 1 = 2 x 3 5 y 3 1
7 x ( 6 x 2 y ) = 7 x ( x 2 y ) 6 1 = 7 x ( x 3 1 y 6 1 ) = 7 x 3 4 y 6 1
7 x 2 ( 6 x y 2 ) = 7 x 2 ( x y 2 ) 6 1 = 7 x 2 ( x 6 1 y 3 1 ) = 7 x 6 13 y 3 1
7 x 2 ( 3 x y 2 ) = 7 x 2 ( x y 2 ) 3 1 = 7 x 2 ( x 3 1 y 3 2 ) = 7 x 3 7 y 3 2
7 x ( 3 x 2 y ) = 7 x ( x 2 y ) 3 1 = 7 x ( x 3 2 y 3 1 ) = 7 x 3 5 y 3 1
Combine like terms Now, let's add the terms:
5 x 3 5 y 3 1 + 2 x 3 5 y 3 1 + 7 x 3 4 y 6 1 + 7 x 6 13 y 3 1 + 7 x 3 7 y 3 2 + 7 x 3 5 y 3 1
Combine the like terms:
( 5 + 2 + 7 ) x 3 5 y 3 1 + 7 x 3 4 y 6 1 + 7 x 6 13 y 3 1 + 7 x 3 7 y 3 2
14 x 3 5 y 3 1 + 7 x 3 4 y 6 1 + 7 x 6 13 y 3 1 + 7 x 3 7 y 3 2
Rewrite with radicals Rewrite the terms with radicals:
14 x 3 5 y 3 1 + 7 x 3 4 y 6 1 + 7 x 6 13 y 3 1 + 7 x 3 7 y 3 2 = 14 x 3 x 2 y + 7 x 6 x 2 y + 7 x 2 6 x y 2 + 7 x 2 3 x y 2
Final Answer Therefore, the sum is:
14 x 3 x 2 y + 7 x 6 x 2 y + 7 x 2 6 x y 2 + 7 x 2 3 x y 2
Examples
Understanding how to simplify expressions with radicals and fractional exponents is crucial in many areas of mathematics and physics. For example, in physics, you might encounter such expressions when dealing with wave functions in quantum mechanics or when calculating the energy levels of a particle in a potential well. In engineering, these concepts are used in signal processing and control systems, where understanding the frequency response of a system often involves manipulating complex expressions with fractional exponents. Simplifying these expressions allows for easier analysis and interpretation of the physical phenomena being modeled.
The expression simplifies to 7 x ( 3 x 2 y ) , which matches option D. This is found by combining and simplifying the given terms using fractional exponents. The final answer is D.
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