Solve $(z-8)^3=-2$ where $z$ is a real number. Simplify your answer as much as possible. (If there is more than one solution, separate them with commas.) $z=\square$
Answer (2)
Take the cube root of both sides: z − 8 = 3 − 2 .
Simplify the cube root: z − 8 = − 3 2 .
Isolate z by adding 8 to both sides: z = 8 − 3 2 .
The solution is 8 − 3 2 .
Explanation
Understanding the Problem We are given the equation ( z − 8 ) 3 = − 2 and we need to solve for z , where z is a real number.
Taking the Cube Root To solve for z , we first take the cube root of both sides of the equation: ( z − 8 ) 3 = − 2
3 ( z − 8 ) 3 = 3 − 2
z − 8 = 3 − 2
Simplifying the Cube Root Since 3 − 2 = − 3 2 , we have z − 8 = − 3 2
Isolating z Now, we add 8 to both sides of the equation to isolate z : z = 8 − 3 2
Final Answer Therefore, the solution is z = 8 − 3 2 .
Examples
Imagine you're designing a cubic container, and you need it to have a volume that is -2 cubic units relative to a standard size. This problem helps you determine the exact adjustment needed to the side length of the container, ensuring it meets the required specifications. Understanding cube roots and algebraic manipulations is crucial in engineering and design for precise calculations and adjustments.
To solve the equation ( z − 8 ) 3 = − 2 , we take the cube root of both sides to find z − 8 = − 3 2 . Adding 8 to both sides gives the final solution z = 8 − 3 2 .
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