Solve for $x$.
$\begin{array}{l}
\sqrt[5]{7 x-4}=1 \\
x=\square
\end{array}$
Answer (1)
Raise both sides of the equation to the power of 5: ( 5 7 x − 4 ) 5 = 1 5 .
Simplify the equation: 7 x − 4 = 1 .
Add 4 to both sides: 7 x = 5 .
Divide by 7 to solve for x : x = 7 5 . The solution is 7 5 .
Explanation
Understanding the Problem We are given the equation 5 7 x − 4 = 1 and asked to solve for x . Our goal is to isolate x on one side of the equation.
Raising Both Sides to the Power of 5 To eliminate the fifth root, we raise both sides of the equation to the power of 5: ( 5 7 x − 4 ) 5 = 1 5
Simplifying the Equation This simplifies to: 7 x − 4 = 1
Adding 4 to Both Sides Next, we add 4 to both sides of the equation to isolate the term with x : 7 x − 4 + 4 = 1 + 4
Simplifying Further This simplifies to: 7 x = 5
Dividing by 7 Finally, we divide both sides by 7 to solve for x : 7 7 x = 7 5
Final Answer Thus, we find that: x = 7 5 Therefore, the solution to the equation is 7 5 .
Examples
Imagine you are designing a circuit where the resistance is determined by the solution to the equation 5 7 x − 4 = 1 . Solving this equation allows you to find the exact value of x needed to achieve the desired resistance in the circuit. This type of problem also appears when modeling physical phenomena where a variable is related to its fifth root. Understanding how to manipulate and solve such equations is crucial in engineering and physics for accurate predictions and designs.
