Solving the Radical Equation: x3 7

When solving the equation where the radical of x cubed equals 7, we are essentially looking for a value of x such that when cubed, it results in 49. This can be represented as:

Problem Presentation

The equation can be written as:

[sqrt{x^3} 7]

To solve for x, we need to simplify and operate step by step.

Solution Process

Let's begin with the given equation:

[sqrt{x^3} 7]

Squaring both sides of the equation to remove the square root gives us:

[sqrt{x^3}^2 7^2]

By simplifying, we get:

[x^3 49]

Step 2: Taking the Cube Root

To solve for x, we need to find the cube root of both sides:

[x sqrt[3]{49}]

The value of (sqrt[3]{49}) is approximately 3.6593, when rounded to four decimal places.

Verification of the Solution

To verify the solution, we can substitute the value back into the original equation:

[sqrt{(3.6593)^3} approx 7]

We can write this out in detail:

First, cube the value 3.6593: [(3.6593)^3 49] Then, take the square root of 49: [sqrt{49} 7]

This confirms that the solution (x sqrt[3]{49}) or approximately 3.6593 is correct.

Final Answer

The solution to the equation (sqrt{x^3} 7) is:

[x sqrt[3]{49} approx 3.6593 , (text{rounded to four decimal places})]

Conclusion

The process of solving the equation involves squaring both sides to eliminate the radical, then taking the cube root to isolate x. This approach is fundamental in understanding how to solve similar radical equations.