Select whether the equation has a solution or not.
[tex]\sqrt[3]{x-5}-2=0[/tex]
A. no roots
B. roots
Answer (2)
Isolate the cube root: 3 x − 5 = 2 .
Cube both sides: x − 5 = 8 .
Solve for x: x = 13 .
The equation has a solution since x = 13 is a real number. The final answer is roots.
Explanation
Understanding the Problem We are given the equation 3 x − 5 − 2 = 0 and asked to determine if it has a solution.
Isolating the Cube Root First, we isolate the cube root term by adding 2 to both sides of the equation: 3 x − 5 = 2
Eliminating the Cube Root Next, we cube both sides of the equation to eliminate the cube root: ( 3 x − 5 ) 3 = 2 3
Simplifying the Equation Simplifying both sides, we get: x − 5 = 8
Solving for x Now, we solve for x by adding 5 to both sides: x = 8 + 5
Finding the Value of x Therefore, we find the value of x : x = 13 Since we found a real value for x , the equation has a solution.
Examples
Imagine you're designing a water tank in the shape of a cube. You need the tank to hold a specific volume of water, and you know the relationship between the side length of the cube and its volume involves a cube root. Solving an equation like the one above helps you determine the exact side length needed to achieve the desired volume. This type of problem arises in various engineering and design scenarios where understanding the relationship between a variable and its cube root is crucial for accurate calculations and practical applications.
The equation 3 x − 5 − 2 = 0 has a solution because we find that x = 13 , a real number. Therefore, the answer is that it has roots.
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