/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 61 \(\sqrt[3]{x-8}+2=0\)... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 61

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

Short Answer

Expert verified
x = 0

Step by step solution

01

- Isolate the Cube Root

Subtract 2 from both sides of the equation to isolate the cube root term: \[\sqrt[3]{x-8} = -2\]
02

- Eliminate the Cube Root

To eliminate the cube root, cube both sides of the equation: \[(\sqrt[3]{x-8})^3 = (-2)^3\]This simplifies to: \[x-8 = -8\]
03

- Solve for x

Add 8 to both sides of the equation to solve for \(x\): \[x = 0\]

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Key Concepts

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

Cube Root Isolation
When tackling a cubic equation, the first goal is often to isolate the cube root term. This step involves using basic algebraic operations to move other terms away from the cube root.
In the given exercise \(\sqrt[3]{x-8}+2=0\), we start by subtracting 2 from both sides of the equation. This gives us the isolated cube root term as follows:
\[\sqrt[3]{x-8} = -2\]
This process simplifies the equation, making it easier to deal with the cube root and progress to the next step.
It's essential to practice isolating terms, as this is a common step in many algebra problems.
Eliminating Cube Roots
Once the cube root term is isolated, you'll want to eliminate the cube root so you can move towards finding the value of \(x\).
Here, we have the equation \[\sqrt[3]{x-8} = -2\]. To eliminate the cube root, we need to cube both sides of the equation. This is because cubing the cube root term and the other side balances the equation and removes the cube root:
\[(\sqrt[3]{x-8})^3 = (-2)^3\]
After performing this operation, the equation simplifies to:
\[x - 8 = -8\]
Without the cube root, the equation now looks simpler and easier to solve.
Remember, cubing is a powerful tool to remove cube roots and should be done carefully to maintain balance in the equation.
Solving for x
Now that the cube root has been eliminated, the final step is to solve for \(x\).
Our simplified equation is \[x - 8 = -8\]. To find the value of \(x\), add 8 to both sides of the equation:
\[x - 8 + 8 = -8 + 8\]
Which further simplifies to:
\[x = 0\]
This tells us that \(x\) equals 0.
It is crucial to check your answer by substituting it back into the original equation to ensure its validity. This not only helps confirm the solution but also reinforces your understanding of the problem.
Practice solving for \(x\) in various equations to build confidence and proficiency in algebra.

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