What values of $c$ and $d$ make the equation true?
$\sqrt[3]{162 x^c y^5}=3 x^2 y\left(\sqrt[3]{6 y^d}\right)$
A. $c=2, d=2$
B. $c=2, d=4$
C. $c=6, d=2$
D. $c=6, d=4$
Answer (2)
Cube both sides of the equation to eliminate the cube roots.
Simplify the equation to isolate the variables.
Equate the exponents of x and y on both sides.
Solve for c and d : c = 6 , d = 2
Explanation
Analyze the problem We are given the equation 3 162 x c y 5 = 3 x 2 y ( 3 6 y d ) and we need to find the values of c and d that make this equation true. Let's start by simplifying the equation.
Cube both sides First, cube both sides of the equation to eliminate the cube roots: ( 3 162 x c y 5 ) 3 = ( 3 x 2 y ( 3 6 y d ) ) 3
Simplify the equation Simplify both sides of the equation: 162 x c y 5 = ( 3 x 2 y ) 3 ( 6 y d ) = 27 x 6 y 3 ( 6 y d ) 162 x c y 5 = 162 x 6 y 3 + d
Divide by 162 Divide both sides by 162: x c y 5 = x 6 y 3 + d
Equate exponents of x Now, equate the exponents of x on both sides: c = 6
Equate exponents of y Equate the exponents of y on both sides: 5 = 3 + d
Solve for d Solve for d :
d = 5 − 3 = 2
Final Answer Therefore, the values of c and d that make the equation true are c = 6 and d = 2 .
Examples
Imagine you are designing a storage container, and the volume is expressed with cube roots and exponents. To optimize the dimensions, you need to simplify the expression and find the correct values for the exponents. This problem demonstrates how to manipulate equations with cube roots and exponents, which is essential in various engineering and design applications, such as determining the dimensions of a container or optimizing the performance of a system.
By solving the equation 3 162 x c y 5 = 3 x 2 y ( 3 6 y d ) , we find that the values for c and d are c = 6 and d = 2 , which corresponds to Option C.
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