Problem 45 Solve each equation. $$ \sqr... [FREE SOLUTION]

Chapter 10: Problem 45

Solve each equation. $$ \sqrt[3]{x-8}+2=0 $$

Short Answer

Expert verified

x = 0

Step by step solution

- Isolate the Cubic Root

First, subtract 2 from both sides of the equation to isolate the cubic root expression. \[ \sqrt[3]{x-8} + 2 - 2 = 0 - 2 \] Simplifies to: \[ \sqrt[3]{x-8} = -2 \]

- Eliminate the Cubic Root

Next, cube both sides of the equation to eliminate the cubic root. \[ ( \sqrt[3]{x-8} )^3 = (-2)^3 \] Simplifies to: \[ x-8 = -8 \]

- Solve for x

Finally, add 8 to both sides of the equation to solve for x. \[ x-8+8 = -8+8 \] Simplifies to: \[ x = 0 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Cubic Root

To solve equations that involve a cubic root, the first important step is isolating the cubic root expression. This means you want to get the term that contains the cubic root by itself on one side of the equation.

Let's look at an example:

\text{Given the equation:} \ \text{\[\sqrt[3]{x-8}+2=0 \]} \

We need to remove the 2 that is added to the cubic root. To do this, we subtract 2 from both sides of the equation:

\[ \sqrt[3]{x-8} + 2 - 2 = 0 - 2 \]

which simplifies to:

\[\sqrt[3]{x-8} = -2 \]

Now, the cubic root expression is isolated on one side of the equation. This process helps in simplifying the next steps as it allows us to focus only on the cubic root term.

Eliminating Cubic Root

Once the cubic root is isolated, the next step is to eliminate the cubic root by cubing both sides of the equation. This means raising both sides of the equation to the power of 3.

From our example, we have:
\[ \sqrt[3]{x-8} = -2 \]
Cubing both sides, we get:

\[ (\sqrt[3]{x-8})^3 = (-2)^3 \]

The cubic root and the cube cancel each other out on the left side, leaving us with:
\[ x-8 = -8 \]
This step is crucial as it removes the cubic root and makes the equation linear, which is easier to solve.

Solving for x

Now that we have a simpler equation, we can solve for x by isolating it. We do this just like solving a basic algebraic equation.

From our reduced equation:

\[ x - 8 = -8 \]

We solve for x by adding 8 to both sides:

\[ x - 8 + 8 = -8 + 8 \]
This simplifies to:
\[ x = 0 \]

By following these steps, we found that the solution to the original equation is 𝛆 = 0.

To summarize, in equations involving cubic roots, always:

Isolate the cubic root term.
Eliminate the cubic root by cubing both sides.
Solve the resulting equation.

This systematic approach makes solving such equations straightforward and manageable.

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91影视

Solve each equation. $$ \sqrt[3]{x-8}+2=0 $$

Short Answer

Step by step solution

- Isolate the Cubic Root

- Eliminate the Cubic Root

- Solve for x

Key Concepts

Isolating Cubic Root

Eliminating Cubic Root

Solving for x

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