Determine the solution for the following equation: $(8 x-8)^{\frac{3}{2}}=64$
A. $x=3$
B. $x=5$
C. $x=13$
D. $x=65$
Answer (2)
Raise both sides of the equation to the power of 3 2 : ( 8 x − 8 ) = 6 4 3 2 .
Calculate 6 4 3 2 : 6 4 3 2 = 16 .
Solve for x : 8 x − 8 = 16 ⇒ 8 x = 24 ⇒ x = 3 .
The solution is 3 .
Explanation
Understanding the Problem We are given the equation ( 8 x − 8 ) 2 3 = 64 and asked to find the value of x that satisfies the equation from the given options: x = 3 , x = 5 , x = 13 , x = 65 .
Raising both sides to the power of 2/3 To solve for x , we first raise both sides of the equation to the power of 3 2 to eliminate the fractional exponent on the left side: (( 8 x − 8 ) 2 3 ) 3 2 = 6 4 3 2
Simplifying the equation Simplifying the equation, we get: 8 x − 8 = 6 4 3 2
Calculating 64^(2/3) Now, we calculate 6 4 3 2 . This can be done by first taking the cube root of 64 and then squaring the result: 6 4 3 2 = ( 6 4 3 1 ) 2 = 4 2 = 16
Substituting the value Substitute the value back into the equation: 8 x − 8 = 16
Adding 8 to both sides Add 8 to both sides of the equation: 8 x = 16 + 8
Simplifying Simplify: 8 x = 24
Dividing by 8 Divide both sides by 8: x = 8 24
Solving for x Solve for x :
x = 3
Final Answer Therefore, the solution to the equation ( 8 x − 8 ) 2 3 = 64 is x = 3 .
Examples
This type of equation can be used to model various physical phenomena, such as the flow rate of a fluid through an opening, where the variable x represents a physical dimension or property of the system. Solving such equations helps engineers and scientists predict and control these phenomena in real-world applications, like designing hydraulic systems or analyzing fluid dynamics.
The solution to the equation ( 8 x − 8 ) 2 3 = 64 is x = 3 . We found this by raising both sides to the power of 3 2 and solving for x . Therefore, the correct answer is option A, x = 3 .
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