Solve the equation, writing the solution as a reduced fraction or as an integer. If there is no real solution, enter "DNE." If there are multiple solutions, separate the solutions with a comma.
Solve:
$\sqrt[3]{-2 r+7}=7$
Answer (2)
Cube both sides of the equation: ( 3 − 2 r + 7 ) 3 = 7 3 .
Simplify the equation: − 2 r + 7 = 343 .
Isolate the term with r : − 2 r = 343 − 7 = 336 .
Solve for r : r = − 2 336 = − 168 .
Explanation
Understanding the Problem We are given the equation 3 − 2 r + 7 = 7 . Our goal is to isolate r and find its value.
Cubing Both Sides To eliminate the cube root, we cube both sides of the equation: ( 3 − 2 r + 7 ) 3 = 7 3 .
Simplifying the Equation Simplifying, we get − 2 r + 7 = 343 .
Isolating the Term with r Next, we isolate the term with r by subtracting 7 from both sides: − 2 r = 343 − 7 , which simplifies to − 2 r = 336 .
Solving for r Finally, we solve for r by dividing both sides by -2: r = − 2 336 , which gives us r = − 168 .
Final Answer Therefore, the solution to the equation is r = − 168 .
Examples
Imagine you are designing a temperature control system for a chemical reaction. The reaction rate is related to temperature by a cube root function. If you know the desired reaction rate (in this case, 7), you can use the equation 3 − 2 r + 7 = 7 to determine the required temperature ( r ) for the reaction. Solving this equation ensures that the reaction proceeds at the optimal rate, which is crucial for efficient production and safety.
To solve the equation 3 − 2 r + 7 = 7 , we cube both sides to get − 2 r + 7 = 343 . After isolating r and solving, we find that r = − 168 .
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