Problem 72 - Algebra 2

Question

Solve each equation. $$ x^{3}+1000=0 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -10\).

Step 1: Rearrange the equation.

The equation given is \(x^{3}+1000=0\). To solve for x, we first need to isolate x^{3} on one side of the equation. We can do this by subtracting 1000 from both sides, which gives us \(x^{3} = -1000\).

Step 2: Extract the cube root.

To solve for x, we now need to take the cube root of both sides of the equation. The cube root of \(x^{3}\) is x, and the cube root of -1000 is -10 (since \(-10 \times -10 \times -10 = -1000\)). So taking the cube root gives us \(x = -10\).

Key Concepts

Cube Root

The concept of cube root can be quite intriguing. The cube root of a number is a value that, when multiplied by itself twice, returns the original number. For instance, the cube root of \(-1000\) is clearly \(-10\), as calculating \(-10 \times -10 \times -10\) results in \(-1000\). It is fundamentally about reversing the operation of cubing a number.

• Cubing involves multiplying a number by itself twice.
• Cube root provides the original number.
• This operation can yield positive or negative results because raising a negative number to an odd power results in a negative product.

The cube root is symbolically represented with an exponent of \(^{1/3}\). Therefore, finding the cube root of \(x^3\) gives us \(x\), a fundamental step when solving cubic equations. In general, the cube root function helps identify roots in cubic equations, just like we saw with transforming \(x^3 = -1000\) into \(x = -10\).

Isolation of Variables

Isolation of variables is a crucial technique in algebra that helps solve equations effectively. The basic idea is to manipulate the equation until the variable of interest is on one side. This often involves simple arithmetic operations like addition, subtraction, multiplication, or division.

• The goal is to "isolate" the variable, usually on the left side.
• This process simplifies the problem, making it easier to discern the variable's value.
• Key operations can include moving terms across the equal sign or combining like terms.

In the context of the exercise \(x^3 + 1000 = 0\), isolating \(x^3\) involved subtracting 1000 from both sides, leading to \(x^3 = -1000\). This step made it clear on which value the subsequent operations should focus. Mastering this skill empowers one to tackle complex equations more confidently.

Solving Equations

Solving equations involves finding the value or values for variables that make the equation true. The methods vary by equation type, but core strategies are applicable across the board. For simple polynomial equations (like our cubic equation), a step-through process leads us to the solution.

• Break down the equation systematically, starting with isolating variables.
• Apply operations, like taking roots, when necessary to simplify further.
• Conduct basic checks by substituting your answer back into the original equation to ensure correctness.

For the equation \(x^3 + 1000 = 0\), solving it boiled down to isolating \(x^3\) and then taking a cube root to find \(x = -10\). Always remember to verify the solution through substitution to confirm that it satisfies the original equation—ensuring accuracy in solving mathematical equations.