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: $\quad \sqrt[3]{-7 l+4}=8$
Solution: $l=$ $\square$
Answer (2)
Cube both sides of the equation: ( 3 − 7 l + 4 ) 3 = 8 3 .
Simplify the equation: − 7 l + 4 = 512 .
Isolate the term with l : − 7 l = 512 − 4 = 508 .
Solve for l : l = − 7 508 = − 7 508 . The final answer is − 7 508 .
Explanation
Understanding the Problem We are given the equation 3 − 7 l + 4 = 8 . Our goal is to solve for l , which means we want to isolate l on one side of the equation.
Cubing Both Sides To get rid of the cube root, we need to cube both sides of the equation. This gives us: ( 3 − 7 l + 4 ) 3 = 8 3
Simplifying the Equation Simplifying both sides, we have: − 7 l + 4 = 512
Isolating the Term with l Next, we want to isolate the term with l . To do this, we subtract 4 from both sides of the equation: − 7 l = 512 − 4
Simplifying Further Simplifying further, we get: − 7 l = 508
Solving for l Now, we solve for l by dividing both sides by -7: l = − 7 508
Final Answer So, the solution is: l = − 7 508 Since 508 is not divisible by 7, the fraction is already in reduced form.
Examples
Imagine you are designing a temperature control system for a chemical reaction. The reaction rate depends on the temperature, and the temperature is related to a control variable l by the equation 3 − 7 l + 4 = 8 . Solving for l allows you to determine the precise setting needed to achieve the desired reaction rate. This type of algebraic manipulation is crucial in engineering and scientific applications where precise control is necessary.
The solution to the equation 3 − 7 l + 4 = 8 is l = − 7 508 , which is already in reduced form.
;
