Problem 71
Question
Solve each equation. $$ 2 x^{3}-16=0 $$
Step-by-Step Solution
Verified Answer
The solution to the equation \(2x^{3} - 16 = 0\) is \(x = 2\).
1Step 1: Rewrite the equation
This step is to simplify the equation by moving the \(-16\) to the right side of the equation. \[2x^{3}=16\], by adding 16 to both sides of the equation.
2Step 2: Solve for \(x^{3}\)
This step is to solve the equation for \(x^{3}\). To do this, divide each side of the equation by 2. \[x^3 = 8\]
3Step 3: Solve for \(x\)
In this step, the cube root is taken on both sides to solve for \(x\). \[ x = \sqrt[3]{8}\]
Key Concepts
Equation SolvingCube RootsSimplifying Equations
Equation Solving
Equation solving helps us find the value of variables that make a mathematical statement true. In the given exercise, we are tasked with solving a cubic equation. To start, we need to work towards isolating the variable, typically represented as "x." This involves rearranging the equation and performing operations on both sides until the variable stands alone on one side of the equation. Let's break these steps down further:
1. **Understand the problem**: The original equation provided is \(2x^{3} - 16 = 0\). This is a cubic equation because it involves \(x^{3}\), which signifies a cube.
2. **Rearrange the terms**: Our primary goal is to isolate \(x^{3}\). We begin by adding 16 to both sides, resulting in \(2x^{3} = 16\). This isolated the cubic term on one side.
3. **Simplify further**: Continue by dividing every term by 2 to find \(x^{3} = 8\). By methodically isolating terms, we make problem-solving easier.
Equation solving provides the solution path to find unknowns in mathematical statements. Simplifying each step brings us closer to the solution.
1. **Understand the problem**: The original equation provided is \(2x^{3} - 16 = 0\). This is a cubic equation because it involves \(x^{3}\), which signifies a cube.
2. **Rearrange the terms**: Our primary goal is to isolate \(x^{3}\). We begin by adding 16 to both sides, resulting in \(2x^{3} = 16\). This isolated the cubic term on one side.
3. **Simplify further**: Continue by dividing every term by 2 to find \(x^{3} = 8\). By methodically isolating terms, we make problem-solving easier.
Equation solving provides the solution path to find unknowns in mathematical statements. Simplifying each step brings us closer to the solution.
Cube Roots
Taking the cube root is crucial when solving cubic equations. Finding the cube root means identifying a number which, when multiplied by itself three times, gives the original number. In this equation, once we have isolated \(x^{3} = 8\), we need to find \(x\) by taking the cube root of 8.
1. **Recognize cube roots**: The cube root of a number, \(N\), is a value that satisfies the equation \(x^{3} = N\). When you cube a number, it becomes raised to the power of three.
2. **Compute \(x\)**: Since \(x^{3} = 8\), compute \(x\) by taking the cube root of 8. This gives \(x = \sqrt[3]{8}\).
3. **Simple calculation**: For many, this calculation might be straightforward if you know your basic cube numbers. Since \(2 imes 2 imes 2 = 8\), it follows that \(x = 2\).
Cube roots revert a number back to its base when cubed, which is necessary for finding accurate solutions in cubic equations like the one in the exercise.
1. **Recognize cube roots**: The cube root of a number, \(N\), is a value that satisfies the equation \(x^{3} = N\). When you cube a number, it becomes raised to the power of three.
2. **Compute \(x\)**: Since \(x^{3} = 8\), compute \(x\) by taking the cube root of 8. This gives \(x = \sqrt[3]{8}\).
3. **Simple calculation**: For many, this calculation might be straightforward if you know your basic cube numbers. Since \(2 imes 2 imes 2 = 8\), it follows that \(x = 2\).
Cube roots revert a number back to its base when cubed, which is necessary for finding accurate solutions in cubic equations like the one in the exercise.
Simplifying Equations
Simplifying equations involves breaking down expressions to their simplest form. This makes it easier to work with and find solutions to problems. In the context of the cubic equation provided, simplification plays a significant role in solving the problem efficiently.
1. **Clear unnecessary terms**: Initially, we had \(2x^{3} - 16 = 0\). By moving "-16" across the equal sign, the equation became simpler: \(2x^{3} = 16\).
2. **Focus on crucial steps**: To get \(x^{3}\) by itself, we divide by 2. This simple operation yields \(x^{3} = 8\). Breaking the problem into smaller, easily solvable parts can make the process less overwhelming.
3. **Ease of calculation**: Lastly, taking the cube root of 8 gives \(x = 2\), a straightforward calculation once the equation is simplified.
Simplifying equations aids in reducing complexity, allowing students to untangle even the most challenging mathematical problems with confidence. Each simplification step brings clarity, making the path to the final answer clear and manageable.
1. **Clear unnecessary terms**: Initially, we had \(2x^{3} - 16 = 0\). By moving "-16" across the equal sign, the equation became simpler: \(2x^{3} = 16\).
2. **Focus on crucial steps**: To get \(x^{3}\) by itself, we divide by 2. This simple operation yields \(x^{3} = 8\). Breaking the problem into smaller, easily solvable parts can make the process less overwhelming.
3. **Ease of calculation**: Lastly, taking the cube root of 8 gives \(x = 2\), a straightforward calculation once the equation is simplified.
Simplifying equations aids in reducing complexity, allowing students to untangle even the most challenging mathematical problems with confidence. Each simplification step brings clarity, making the path to the final answer clear and manageable.
Other exercises in this chapter
Problem 71
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