Problem 45 - College Algebra with Modeling and Visualization

Question

Solve the equation. Check your answers. $$ \sqrt[3]{z+1}=-3 $$

Step-by-Step Solution

Answer

The solution is \( z = -28 \).

Step 1: Understand the Equation

The equation we need to solve is \( \sqrt[3]{z+1} = -3 \). This equation involves a cube root on the left side, and our goal is to isolate \( z \).

Step 2: Lift the Cube Root

To eliminate the cube root, we need to cube both sides of the equation. This gives us: \((\sqrt[3]{z+1})^3 = (-3)^3\). Cubing cancels out the cube root on the left side.

Step 3: Simplify the Equation

After cubing both sides, we get: \( z+1 = -27 \). Now the equation is simplified to a linear form.

Step 4: Solve for z

To find \( z \), subtract 1 from both sides: \( z = -27 - 1 \). Thus, \( z = -28 \).

Step 5: Check the Solution

Substitute \( z = -28 \) back into the original equation to verify: \( \sqrt[3]{-28+1} = \sqrt[3]{-27} = -3 \), which matches the right side of the original equation.

Key Concepts

Cube Root

A cube root is a number that, when multiplied by itself twice, produces the original number. It is represented by the symbol \(\sqrt[3]{x}\). For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). In our exercise, we are working with the cube root of an expression, specifically \(\sqrt[3]{z+1}\).

The key property of cube roots is that they allow us to "reverse" the process of cubing a number. When we cube both sides of an equation containing a cube root, we remove or "lift" the cube root, leaving us with a simpler equation to solve. This method is essential when the goal is to isolate and solve for a variable like \(z\).

Remember, taking the cube root can produce both positive and negative results since cubing a negative number also gives a negative result. In our problem, when you see \(\sqrt[3]{z+1} = -3\), it indicates that the cube of the result is negative, which is expected because \(z+1\) might be a negative number as well.

Linear Equation

A linear equation is an equation between two variables that gives a straight line when plotted on a graph. These equations are typically in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

In the exercise, once we remove the cube root by cubing both sides, we simplify the original equation \(\sqrt[3]{z+1} = -3\) to a linear form: \(z+1 = -27\).

Linear equations are straightforward to solve, as they involve simple arithmetic operations to isolate the variable. In our case, \(z\) is isolated by subtracting 1 from both sides of the equation, resulting in \(z = -28\).

This transformation from a more complex equation with a cube root to a simple linear equation illustrates how understanding the properties of different kinds of equations can simplify solving them.

Equation Solving

Solving equations involves finding the value of the variable that makes the equation true. This process often involves several steps, like simplification, isolation of the variable, and arithmetic operations.

Let's consider the example at hand: starting with \(\sqrt[3]{z+1}=-3\). Solving this involves:

• Removing the cube root by cubing both sides, resulting in \(z+1 = -27\).
• Isolating \(z\) through subtraction: \(z = -27 - 1\).
• Simplifying the expression to find \(z = -28\).

Each of these steps brings us closer to the solution by reducing complexity and applying operations that preserve equality, leading us from a cube root equation to a simple arithmetic equation.

Therefore, solving equations is about taking logical and procedural steps to arrive at the answer, preserving the balance of the equation at each step.

Verification of Solutions

After finding a solution to an equation, it is essential to verify that this solution satisfies the original equation. Verification is a crucial step in problem-solving as it confirms the correctness of your solution.

In our problem, after calculating \(z = -28\), we substitute it back into the original equation to see if it holds true:

• Check: \(\sqrt[3]{-28+1} = \sqrt[3]{-27}\)
• Result: \(-3 = -3\)

As both sides of the equation are equal, this confirms that our solution is correct.

Verification ensures that the solution does not just theoretically satisfy the steps but also accurately solves the equation under given conditions. This process helps catch any mistakes that might have occurred during solving, providing a reliable way to ensure understanding and accuracy.