Type the correct answer in the box. Use numerals instead of words. What is the solution to this equation? [tex]$7-\sqrt[3]{2-x}=12$[/tex] The solution is [tex]$x=$[/tex] $\square$.
Answer (2)
Isolate the cube root term: − 3 2 − x = 5 .
Multiply both sides by -1: 3 2 − x = − 5 .
Cube both sides: 2 − x = − 125 .
Solve for x : x = 127 .
127
Explanation
Understanding the Problem We are given the equation 7 − 3 2 − x = 12 and we need to find the value of x that satisfies this equation.
Isolating the Cube Root First, we want to isolate the cube root term. To do this, we subtract 7 from both sides of the equation: 7 − 3 2 − x − 7 = 12 − 7
Simplifying the Equation Simplifying both sides, we get: − 3 2 − x = 5
Removing the Negative Sign Next, we multiply both sides by -1 to get rid of the negative sign on the cube root: ( − 1 ) × − 3 2 − x = ( − 1 ) × 5
3 2 − x = − 5
Cubing Both Sides Now, we cube both sides of the equation to eliminate the cube root: ( 3 2 − x ) 3 = ( − 5 ) 3
Simplifying Further Simplifying both sides, we have: 2 − x = − 125
Isolating x To isolate x , we subtract 2 from both sides: 2 − x − 2 = − 125 − 2
− x = − 127
Solving for x Finally, we multiply both sides by -1 to solve for x : ( − 1 ) × − x = ( − 1 ) × − 127
x = 127
Final Answer Therefore, the solution to the equation is x = 127 .
Examples
Imagine you're designing a temperature control system where the temperature is modeled by a cube root function. Solving equations like this helps you determine the input needed to achieve a specific desired temperature. This kind of problem is also fundamental in many engineering applications, such as designing systems with feedback loops or analyzing the behavior of certain types of sensors. Understanding how to manipulate and solve equations involving cube roots is a key skill in these fields.
The solution to the equation 7 − 3 2 − x = 12 is x = 127 . We isolate the cube root, remove the negative sign, cube both sides, and solve for x step-by-step. This method confirms the value of x satisfies the given equation.
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