Problem 61 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{3}=27\)

Step-by-Step Solution

Answer

The real solution is \(x = 3\).

Step 1: Identify the type of equation

The given equation is a cubic equation because it involves a variable raised to the power of 3, i.e., \(x^3 = 27\).

Step 2: Isolate the variable

To solve for \(x\), we need to isolate it. Since we are dealing with a power of 3, we need to take the cube root of both sides of the equation: \(x = \sqrt[3]{27}\).

Step 3: Evaluate the cube root

Calculate the cube root of 27. Since \(3^3 = 27\), we know that \(\sqrt[3]{27} = 3\).

Step 4: Find all real solutions

For the given cubic equation \(x^3 = 27\), the only real solution is \(x = 3\) since there are no negative or non-real components involved in the solution.

Key Concepts

Real Solutions

When solving a cubic equation, like \(x^3 = 27\), we aim to find the real solutions. Real solutions are the values of \(x\) that make the equation true, using numbers from the set of real numbers. These are numbers without any imaginary parts.

In our example \(x^3 = 27\), the goal is to discover the real value of \(x\) that satisfies the equation. A cubic equation can potentially have three real solutions, but depending on the equation, there might be fewer. Specifically, for \(x^3 = 27\), we merely have one real solution: \(x = 3\). Here, the solution is straightforward because 27 is a perfect cube.

Perfect cubes are numbers like 1, 8, 27, etc., that have an integer as their cube root. When dealing with other cubic equations, the process might involve more steps if the equation is not as neat as this one.

Isolating the Variable

To isolate the variable in equations like \(x^3 = 27\), you aim to get \(x\) alone on one side of the equation. This step is crucial because it sets the stage for determining the solution. Think of it as peeling away layers to discover \(x\).

In the equation \(x^3 = 27\), since \(x\) is raised to the third power, you can isolate \(x\) by reversing that operation. You do this by taking the cube root of both sides. This step transforms the equation into a more manageable form: \(x = \sqrt[3]{27}\).

• Isolating helps simplify the equation.
• It provides clarity, making it easier to solve.

By isolating \(x\), you directly prepare for the next problem-solving step. Always remember, isolating \(x\) is essential in solving not just cubic equations, but most algebraic equations.

Cube Root Calculation

Calculating the cube root is the essential final step to finding the real solution in an equation like \(x^3 = 27\). Cube root calculation involves finding a number that, when multiplied by itself twice, gives the original number.

For the equation \(x^3 = 27\), we calculate the cube root using \(x = \sqrt[3]{27}\). Here, you are looking for a number \(x\) that satisfies the condition \(x \times x \times x = 27\).

• In this instance, \(3^3 = 27\), hence \(\sqrt[3]{27} = 3\).
• This shows the simplicity when the number is a perfect cube.

Cube roots are critical in solving cubic equations. While taking a cube root might seem daunting at first, especially with larger numbers, knowing how to handle perfect cubes simplifies the process. For more complex numbers, technology, like calculators, can aid in determining non-integer cube roots.