Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation by hand. Do not use a calculator. $$\sqrt[3]{x+1}=-3$$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Cube Root

Identify that the cube root expression is already isolated in the equation \( \sqrt[3]{x+1} = -3 \).

Step 2: Eliminate the Cube Root

To eliminate the cube root, cube both sides of the equation, resulting in \((\sqrt[3]{x+1})^3 = (-3)^3 \), which simplifies to \( x+1 = -27 \).

Step 3: Solve for x

Subtract 1 from both sides of the equation to solve for \( x \): \( x = -27 - 1 \), which simplifies to \( x = -28 \).

Step 4: Verify the Solution

Substitute \( x = -28 \) back into the original equation to verify: \( \sqrt[3]{-28+1} = \sqrt[3]{-27} = -3 \), confirming the solution is correct.

Key Concepts

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In mathematical terms, if you have a number, say \( x \), and its cube root is \( y \), then \( y^3 = x \). Cube roots differ from square roots in that they can yield negative results. This means \( \sqrt[3]{-27} \) is \( -3 \) because \( (-3) \times (-3) \times (-3) = -27 \).

• Non-negative Numbers: Always have a single cube root that is either positive or zero.
• Negative Numbers: Typically yield a single cube root that is negative.

Understanding cube roots is essential because they frequently appear in algebraic equations and real-world problems.
When you see \( \sqrt[3]{x+1} = -3 \), you are trying to find a number that, when added to 1 and then cubed, equals \( -27 \). By isolating and eliminating the cube root through cubing both sides, you move a step closer to solving for \( x \).

Isolating Variables

Isolating a variable is one of the fundamental steps in solving an equation. It involves rearranging the equation so that the variable of interest is by itself on one side of the equation. In the given exercise, the expression \( \sqrt[3]{x+1} = -3 \) is already isolated because the cube root and \( x+1 \) is on its own.

• Purpose: Simplifies the problem and helps to focus directly on solving for the desired variable.
• Technique: Use algebraic operations such as addition, subtraction, multiplication, or division to shift other terms away from your variable.

The isolation step is crucial since it sets up your equation in a simple form, making it easier to apply further algebraic manipulations. This preparation step helps in effectively focusing on the variable you need to solve for.

Algebraic Manipulation

Algebraic manipulation is crucial for transforming equations in a way that allows us to solve for unknowns. In our problem, this involves eliminating the cube root by cubing both sides, turning \( (\sqrt[3]{x+1})^3 = (-3)^3 \) into a more manageable form: \( x+1 = -27 \). This is called raising both sides to the power of three, an inverse operation of taking the cube root.

• Basic Operations: Include addition, subtraction, multiplication, and division to manipulate equations.
• Exponent Rules: Used to simplify expressions involving powers and roots.

Manipulating the equation appropriately allows one to isolate and solve for \( x \). The goal is to get \( x \) by itself, making it easier to proceed with simple arithmetic operations like subtraction or addition.

Verification of Solutions

Verifying a solution helps ensure that our mathematical work is correct. After finding \( x = -28 \), substitute it back into the original equation to make sure it holds true. Specifically, check if \( \sqrt[3]{-28+1} = \sqrt[3]{-27} = -3 \).

• Why Verify? Prevents mistakes by confirming the solution satisfies the original equation.
• Method: Plug the solution back into the original equation and perform the operations to see if both sides are equal.

Verification benefits us by eliminating errors, reinforcing the correctness of our approach, and ensuring that every step logically leads to a valid solution. By verifying, we gain confidence in the answer provided and the methods used to achieve it.