Problem 80 - Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt[3]{x+4}=-1 $$

Step-by-Step Solution

Answer

The solution is \(x = -5\). There are no extraneous solutions.

Step 1: Understand the Equation

The equation given is \( \sqrt[3]{x+4} = -1 \). This denotes the cube root of \( x+4 \) is equal to \(-1\).

Step 2: Eliminate the Cube Root

To remove the cube root, we cube both sides of the equation. This gives us \((\sqrt[3]{x+4})^3 = (-1)^3\). Simplifying, we get \(x + 4 = -1\).

Step 3: Solve for x

Subtract 4 from both sides of the equation \(x + 4 = -1\). This results in \(x = -1 - 4\), which simplifies to \(x = -5\).

Step 4: Check for Extraneous Solutions

Plug \(x = -5\) back into the original equation to verify it is a solution: \(\sqrt[3]{-5+4} = \sqrt[3]{-1} = -1\). Since both sides of the equation are equal, \(x = -5\) is not extraneous and is a valid solution.

Key Concepts

Eliminating Cube Roots

To solve cube root equations like \( \sqrt[3]{x+4} = -1 \), the first step is to eliminate the cube root. Cube roots can be simplified by raising both sides of the equation to the power of three.

This process works as follows: when you cube \( \sqrt[3]{x+4} \), the cube and the cube root cancel each other out, leaving you with just \( x + 4 \). For the right side, \( (-1)^3 \) simplifies to \(-1\).

This transformation gives us a more straightforward equation to solve: \( x + 4 = -1 \). By removing the cube root, we've simplified the problem into a basic algebraic equation where the value of \( x \) can be easily isolated.

Extraneous Solutions

In mathematics, particularly when dealing with equations involving roots, there's the possibility of extraneous solutions. These are solutions that emerge from the algebraic manipulations but do not satisfy the original equation.

When we cubed both sides of \( \sqrt[3]{x+4} = -1 \), we might introduce solutions that don't actually work in the original context due to losing constraints set by the cube root. However, in this particular case, after finding \( x = -5 \), checking back into the original equation confirmed it was not extraneous.

Always remember to verify your solutions to eliminate the possibility of extraneous results, especially when dealing with roots or fractional powers.

Checking Solutions

After solving for \( x \), it's crucial to verify that the solutions fit within the original equation. This process helps confirm the validity of the proposed solution and ensures no extraneous results are present.

For the equation \( \sqrt[3]{x+4} = -1 \), we found \( x = -5 \). When substituted back into the original equation, substituting gives \( \sqrt[3]{-5+4} = \sqrt[3]{-1} \), which simplifies to \(-1\), matching the right side of the equation.

This confirmation step is essential as it provides assurance that \( x = -5 \) truly satisfies the initial problem constraints. Therefore, always remember to perform this check to ensure your solutions are accurate and reliable.