What is the following sum?
$\begin{array}{l}
5 x(\sqrt[3]{x^2 y})+2(\sqrt[3]{x^5 y}) \\
07 x(\sqrt[6]{x^2 y}) \\
07 x^2(\sqrt[6]{x y^2}) \\
07 x^2(\sqrt[3]{x y^2}) \\
07 x(\sqrt[3]{x^2 y})
\end{array}$
Answer (1)
Rewrite each term using fractional exponents.
Combine the coefficients of the like terms.
The final sum is 14 x 5/3 y 1/3 + 7 x 4/3 y 1/6 + 7 x 13/6 y 1/3 + 7 x 7/3 y 2/3 .
Explanation
Understanding the Problem 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 )
We need to simplify each term and then add them together.
Rewriting with Fractional Exponents First, let's rewrite each term using fractional exponents:
5 x ( x 2 y ) 1/3 = 5 x 1 x 2/3 y 1/3 = 5 x 5/3 y 1/3 2 ( x 5 y ) 1/3 = 2 x 5/3 y 1/3 7 x ( x 2 y ) 1/6 = 7 x 1 x 2/6 y 1/6 = 7 x 4/3 y 1/6 7 x 2 ( x y 2 ) 1/6 = 7 x 2 x 1/6 y 2/6 = 7 x 13/6 y 1/3 7 x 2 ( x y 2 ) 1/3 = 7 x 2 x 1/3 y 2/3 = 7 x 7/3 y 2/3 7 x ( x 2 y ) 1/3 = 7 x 1 x 2/3 y 1/3 = 7 x 5/3 y 1/3
Combining Like Terms Now, let's add the terms together:
5 x 5/3 y 1/3 + 2 x 5/3 y 1/3 + 7 x 4/3 y 1/6 + 7 x 13/6 y 1/3 + 7 x 7/3 y 2/3 + 7 x 5/3 y 1/3
Combine the like terms:
( 5 + 2 + 7 ) x 5/3 y 1/3 + 7 x 4/3 y 1/6 + 7 x 13/6 y 1/3 + 7 x 7/3 y 2/3
14 x 5/3 y 1/3 + 7 x 4/3 y 1/6 + 7 x 13/6 y 1/3 + 7 x 7/3 y 2/3
Final Sum So the sum is:
14 x 5/3 y 1/3 + 7 x 4/3 y 1/6 + 7 x 13/6 y 1/3 + 7 x 7/3 y 2/3
Examples
Understanding how to simplify and combine terms with fractional exponents is crucial in many areas of science and engineering. For example, in fluid dynamics, you might encounter expressions involving fractional powers when dealing with flow rates or pressure gradients. Simplifying these expressions allows engineers to make accurate predictions about the behavior of fluids in various systems, such as pipelines or aircraft wings. Also, in thermodynamics, fractional exponents appear in equations of state for real gases, and simplifying these expressions helps scientists to understand and predict the behavior of gases under different conditions.
