What is the following sum?
[tex]$5 x\left(\sqrt[3]{x^2 y}\right)+2\left(\sqrt[3]{x^5 y}\right)$[/tex]
Answer (2)
The expression 5 x ( 3 x 2 y ) + 2 ( 3 x 5 y ) simplifies to 7 3 x 5 y . This is achieved by rewriting each term using fractional exponents, combining like terms, and converting back to radical form.
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Rewrite each term using fractional exponents.
Combine like terms.
Convert the simplified expression back to radical form.
The final 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
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 ) .
To simplify the expression, we will rewrite each term using fractional exponents and then combine like terms.
Rewriting 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
Combining Like Terms Now, let's combine the like 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 = ( 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
Converting Back to Radical Form Rewriting the simplified expression using 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 and combine radical expressions is useful in various fields, such as physics and engineering, where complex equations often involve radicals. For instance, when calculating the impedance of an electrical circuit or determining the stress on a structural component, engineers often encounter expressions with radicals that need simplification. Simplifying these expressions allows for easier calculations and a better understanding of the underlying relationships between variables. This skill is also crucial in computer graphics for rendering complex shapes and textures efficiently.
