Solve the equation with rational exponents. [tex]$(x-1)^{2 / 3}=4$[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is { }. (Simplify your answer. Use a comma to separate answers as needed.)
B. The solution set is the empty set.
Answer (2)
Raise both sides to the power of 2 3 : ( x − 1 ) = ± 4 2 3 .
Evaluate 4 2 3 : 4 2 3 = 8 .
Solve for x in both cases: x = 1 ± 8 .
The solutions are x = 9 and x = − 7 , so the solution set is { − 7 , 9 } .
Explanation
Understanding the Problem We are given the equation ( x − 1 ) 2/3 = 4 . Our goal is to solve for x . Since the exponent is a rational number, we need to be careful when taking roots to ensure we consider all possible solutions.
Raising Both Sides to the Power of 3/2 To eliminate the rational exponent, we raise both sides of the equation to the power of 2 3 . This gives us (( x − 1 ) 2/3 ) 3/2 = 4 3/2 Note that since the numerator of the exponent is even, we must consider both positive and negative roots.
Simplifying the Equation Simplifying the left side, we get x − 1 = ± 4 3/2 Now we evaluate the right side. We have 4 3/2 = ( 4 1/2 ) 3 = 2 3 = 8 . So, we have x − 1 = ± 8
Solving for x Now we solve for x . We have two cases: Case 1: x − 1 = 8 , which gives x = 9 .
Case 2: x − 1 = − 8 , which gives x = − 7 .
Checking the Solutions We need to check both solutions in the original equation to make sure they are valid. For x = 9 , we have ( 9 − 1 ) 2/3 = 8 2/3 = ( 8 1/3 ) 2 = 2 2 = 4 . This solution is valid. For x = − 7 , we have ( − 7 − 1 ) 2/3 = ( − 8 ) 2/3 = (( − 8 ) 1/3 ) 2 = ( − 2 ) 2 = 4 . This solution is also valid.
Final Answer Therefore, the solution set is { − 7 , 9 } .
Examples
Rational exponents appear in various contexts, such as calculating growth rates or solving engineering problems. For example, if the area of a square grows by a factor of 4 over a certain time period, the side length grows by a factor of 4 1/2 = 2 . Similarly, in electrical engineering, the impedance of a circuit can involve rational exponents. Understanding how to solve equations with rational exponents is crucial for modeling and analyzing these real-world phenomena.
The solution set for the equation ( x − 1 ) 2/3 = 4 is { -7, 9 }. Both solutions satisfy the original equation when checked. Therefore, the correct choice is A: The solution set is { -7, 9 }.
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