91Ó°ÊÓ

A cube root is a number that, when multiplied by itself three times, gives the original number. In algebra, the cube root of a number is often represented by the radical symbol with a little 3 inside it, like this: \(\sqrt[3]{x}\). For example, \(\sqrt[3]{27} = 3\) because \(3 \times 3 \times 3 = 27\). Cube roots are important for solving cubic equations. These are equations where the variable is raised to the power of three.
In the exercise given, we started with \(\sqrt[3]{r+1} + 1 = 0\). Our first goal was to isolate the cube root expression. By subtracting 1 from both sides, we turned it into \(\sqrt[3]{r+1} = -1\). Understanding cube roots helps simplify and solve such equations.

Isolating variables means getting the variable by itself on one side of the equation. It's a critical step in solving equations of any kind. By isolating the variable, we can better understand how changes in one part of the equation affect the other parts.
In our exercise, we started with the equation \(\sqrt[3]{r+1} + 1 = 0\). To isolate the cube root expression, we subtracted 1 from both sides. This led us to \(\sqrt[3]{r+1} = -1\). Now, the variable inside the cube root is more easily accessible for further simplification.

• First, look for any constant terms added or subtracted from the term with the variable.
• Perform the opposite operation to both sides of the equation.
• Repeat as necessary until the variable term is isolated.

Once you've isolated the variable, the next step is to solve for it. This often involves performing operations to both sides of the equation until the variable stands alone.
After isolating the cube root in our exercise, we had \(\sqrt[3]{r+1} = -1\). To solve for \(r\), we needed to remove the cube root by cubing both sides. This gave us \( (\sqrt[3]{r+1})^3 = (-1)^3 \). Simplifying further, we got \( r + 1 = -1 \).
Finally, we isolated \( r \) by subtracting 1 from both sides: \( r = -2 \).

• Identify the operation removing or adding any constants with the variable.
• Perform opposite operations to both sides to isolate the variable.
• Ensure you simplify all expressions post-operations to get the variable alone.