Solve For \[$ X \$\] In The Equation:$\[ \sqrt[3]{x} + 27 = 30 \\]

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Introduction

In this article, we will delve into solving for x in the equation $\sqrt[3]{x} + 27 = 30$. This equation involves a cube root, which can be solved using algebraic manipulations. We will break down the solution step by step, providing a clear and concise explanation of each step.

Understanding the Equation

The given equation is $\sqrt[3]{x} + 27 = 30$. To solve for x, we need to isolate the cube root term. The first step is to subtract 27 from both sides of the equation.

Subtracting 27 from Both Sides

$\sqrt[3]{x} + 27 - 27 = 30 - 27$

$\sqrt[3]{x} = 3$

Understanding the Cube Root

The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In this case, we have $\sqrt[3]{x} = 3$. This means that $x = 3^3$.

Cubing Both Sides

To find the value of x, we need to cube both sides of the equation.

$x = 3^3$

$x = 27$

Verifying the Solution

To verify the solution, we can substitute x back into the original equation.

$\sqrt[3]{x} + 27 = 30$

$\sqrt[3]{27} + 27 = 30$

$3 + 27 = 30$

$30 = 30$

The solution checks out, and we have successfully solved for x in the equation $\sqrt[3]{x} + 27 = 30$.

Conclusion

Solving for x in the equation $\sqrt[3]{x} + 27 = 30$ involves isolating the cube root term and then cubing both sides of the equation. By following these steps, we can find the value of x and verify the solution. This equation is a great example of how algebraic manipulations can be used to solve equations involving cube roots.

Real-World Applications

Cube roots have many real-world applications, including:

• Geometry: Cube roots are used to calculate the volume of cubes and rectangular prisms.
• Physics: Cube roots are used to calculate the energy of particles and the frequency of waves.
• Engineering: Cube roots are used to calculate the stress and strain on materials.

Tips and Tricks

• Use algebraic manipulations: Algebraic manipulations are a powerful tool for solving equations involving cube roots.
• Check your work: Always verify your solution by substituting the value back into the original equation.
• Practice, practice, practice: The more you practice solving equations involving cube roots, the more comfortable you will become with the process.

Common Mistakes

• Forgetting to cube both sides: Failing to cube both sides of the equation can lead to an incorrect solution.
• Not verifying the solution: Failing to verify the solution can lead to an incorrect answer.
• Not using algebraic manipulations: Failing to use algebraic manipulations can make the solution more difficult and time-consuming.

Conclusion

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Introduction

In our previous article, we solved for x in the equation $\sqrt[3]{x} + 27 = 30$. In this article, we will answer some common questions that readers may have about solving equations involving cube roots.

Q: What is a cube root?

A: A cube root is a value that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because $3 \times 3 \times 3 = 27$.

Q: How do I solve an equation involving a cube root?

A: To solve an equation involving a cube root, you need to isolate the cube root term and then cube both sides of the equation. This will allow you to find the value of x.

Q: What if I have a negative number under the cube root?

A: If you have a negative number under the cube root, you will need to use the imaginary unit, i, to solve the equation. For example, if you have $\sqrt[3]{-27} = x$, you can cube both sides of the equation to get $-27 = x^3$. Then, you can take the cube root of both sides to get $x = \sqrt[3]{-27} = -3i$.

Q: Can I use a calculator to solve equations involving cube roots?

A: Yes, you can use a calculator to solve equations involving cube roots. However, it's always a good idea to verify your solution by substituting the value back into the original equation.

Q: What if I have a fraction under the cube root?

A: If you have a fraction under the cube root, you will need to rationalize the denominator before you can cube both sides of the equation. For example, if you have $\sqrt[3]{\frac{1}{27}} = x$, you can cube both sides of the equation to get $\frac{1}{27} = x^3$. Then, you can take the cube root of both sides to get $x = \sqrt[3]{\frac{1}{27}} = \frac{1}{3}$.

Q: Can I use algebraic manipulations to solve equations involving cube roots?

A: Yes, you can use algebraic manipulations to solve equations involving cube roots. In fact, algebraic manipulations are a powerful tool for solving equations involving cube roots.

Q: What are some common mistakes to avoid when solving equations involving cube roots?

A: Some common mistakes to avoid when solving equations involving cube roots include:

• Forgetting to cube both sides of the equation
• Not verifying the solution by substituting the value back into the original equation
• Not using algebraic manipulations to simplify the equation

Conclusion

Solving for x in the equation $\sqrt[3]{x} + 27 = 30$ involves isolating the cube root term and then cubing both sides of the equation. By following these steps, we can find the value of x and verify the solution. This equation is a great example of how algebraic manipulations can be used to solve equations involving cube roots.

Additional Resources

• Cube Root Calculator: A calculator that can be used to find the cube root of a number.
• Algebraic Manipulations: A tutorial on how to use algebraic manipulations to solve equations involving cube roots.
• Cube Root Formula: A formula for finding the cube root of a number.

Conclusion

Solving for x in the equation $\sqrt[3]{x} + 27 = 30$ involves isolating the cube root term and then cubing both sides of the equation. By following these steps, we can find the value of x and verify the solution. This equation is a great example of how algebraic manipulations can be used to solve equations involving cube roots.