Problem 113 - Elementary and Intermediate Algebra

Question

What is wrong with the work shown below? $$ \begin{aligned} &\sqrt[3]{x+1}-3=8\\\ &\sqrt[4]{x+x}=11\\\ &\begin{aligned} (\sqrt[3]{x+1})^{3} &=11 \\ x+1 &=11 \\ x &=10 \end{aligned} \end{aligned} $$

Step-by-Step Solution

Answer

The mistake is in solving the original equation; the correct solution for x is 1330.

Step 1: Identify the equation that contains the mistake

Observe the given equations carefully:1. \( \sqrt[3]{x+1}-3=8\ 2. \sqrt[4]{x+x}=11 \ 3. (\sqrt[3]{x+1})^{3} = 11 \) leads to \( x+1 = 11 \) and \( x = 10 \).The mistake is in the manipulation of equation 1, which should have been algebraically manipulated and not the equation 3 as shown.

Step 2: Analyze the first equation

Equation 1 is \( \sqrt[3]{x+1}-3=8 \). To isolate the cube root, 3 should be added to both sides:\[ \sqrt[3]{x+1} = 8 + 3 = 11 \] .Further manipulation is necessary to solve for \( x \). The given solution mistakenly skips these steps.

Step 3: Properly solve the first equation

Continue solving \( \sqrt[3]{x+1} = 11 \) by cubing both sides:\[ (\sqrt[3]{x+1})^3 = 11^3 \] This implies:\[ x + 1 = 11^3 \] Calculate \( 11^3 \) which equals 1331:\[ x + 1 = 1331 \] Solve for \( x \) by subtracting 1 from both sides:\[ x = 1330 \].

Step 4: Validate your solution

Ensure that the solution is consistent:Substitute \( x = 1330 \) back into the original equation \( \sqrt[3]{x+1} = 11 \):\[ \sqrt[3]{1330+1} = \sqrt[3]{1331} = 11 \]This confirms that our corrected solution is valid.

Key Concepts

Cube Root Manipulation

To solve an equation involving a cube root, it's important to isolate the term with the cube root before eliminating it. For instance, in the equation \( \sqrt[3]{x+1} - 3 = 8 \), the term \( \sqrt[3]{x+1} \) is paired with a subtraction, which must be addressed first. Start by adding 3 to both sides to simplify the equation to \( \sqrt[3]{x+1} = 11 \). This step ensures that the cube root is isolated correctly for further manipulation.

Once the cube root is isolated, you can remove it by cubing both sides. Cubing cancels out the cube root, as the operations are inverse. Thus, \( (\sqrt[3]{x+1})^3 = 11^3 \) simplifies the equation to \( x+1 = 11^3 \). Always remember that careful manipulation is crucial to retaining the integrity of the equation.

Finally, solve \( x+1 = 1331 \) by simply subtracting 1 from both sides, which results in \( x = 1330 \). The key in cube root manipulation is to perform logical steps sequentially and stay consistent with basic algebraic operations to reach the correct solution.

Equation Solving Errors

Equation solving errors often stem from a misunderstanding of algebraic manipulation. Errors can easily occur when changing the structure of an equation without maintaining equivalence. In the exercise, the mistake was manipulating an incorrect form of the equation and simplifying in a way that broke its structure.

The solution approach initially skipped isolating the cube root term and incorrectly cubed a misaligned part of the equation. Always aim to solve based on precise arrangements: first isolate, then activate inverse operations, and finally simplify.

To prevent similar errors, follow these tips:

• Identify and isolate the variable or root term within an equation.
• Ensure each step of manipulation maintains the equation’s equivalence.
• Check completed steps for consistency and recognize when operations need reversal.

By adhering to this systematic process, solving equations becomes more reliable, reducing chances for errors.

Mathematical Validation

After solving an equation, it’s essential to confirm the solution through mathematical validation. This process verifies that the solution satisfies the original equation. It involves plugging the solution back into the initial equation to check for consistency.

For example, after solving \( \sqrt[3]{x+1} = 11 \) and finding \( x = 1330 \), substituting \( x \) back into the equation ensures its accuracy: \( \sqrt[3]{1330+1} = \sqrt[3]{1331} = 11 \). This step guarantees no oversight occurred during manipulation and calculation.

Validation has several benefits:

• Confirms the solution is free of errors.
• Ensures the problem-solving approach is correct.
• Highlights the importance of returning to the original equation as a checkpoint.

By always including a validation step, not only is confidence in the solution increased, but practice in verifying results enhances overall problem-solving skills.