What Is The Value Of $z$?$3 \sqrt{z} - 8 = \sqrt{z}$

Introduction

In this article, we will delve into the world of algebra and explore a simple yet intriguing equation: $3 \sqrt{z} - 8 = \sqrt{z}$. Our goal is to solve for the value of $z$, a variable that has been shrouded in mystery. By following a series of logical steps, we will uncover the solution and gain a deeper understanding of the underlying mathematics.

Understanding the Equation

The given equation is $3 \sqrt{z} - 8 = \sqrt{z}$. At first glance, it may seem daunting, but let's break it down into smaller, more manageable parts. We have two terms on the left-hand side: $3 \sqrt{z}$ and $-8$. The right-hand side is simply $\sqrt{z}$.

Isolating the Square Root Term

To solve for $z$, we need to isolate the square root term. Let's start by adding $8$ to both sides of the equation:

$$3 \sqrt{z} = \sqrt{z} + 8$$

This simplifies the equation and allows us to focus on the square root term.

Simplifying the Equation

Next, let's subtract $\sqrt{z}$ from both sides of the equation:

$$3 \sqrt{z} - \sqrt{z} = 8$$

This simplifies to:

$$2 \sqrt{z} = 8$$

Solving for z

Now that we have isolated the square root term, we can solve for $z$. Let's divide both sides of the equation by $2$:

$$\sqrt{z} = 4$$

To solve for $z$, we need to square both sides of the equation:

$$z = 4^2$$

$$z = 16$$

Conclusion

In this article, we have solved for the value of $z$ using a simple yet effective approach. By isolating the square root term, simplifying the equation, and solving for $z$, we have uncovered the solution: $z = 16$. This demonstrates the power of algebra and the importance of breaking down complex equations into smaller, more manageable parts.

Real-World Applications

The concept of solving for $z$ has numerous real-world applications. In physics, for example, the value of $z$ can represent the distance between two objects. In engineering, it can represent the length of a wire or cable. By understanding how to solve for $z$, we can apply this knowledge to a wide range of problems and scenarios.

Common Mistakes to Avoid

When solving for $z$, it's essential to avoid common mistakes. One mistake is to forget to isolate the square root term, which can lead to incorrect solutions. Another mistake is to neglect to square both sides of the equation, which can result in an incorrect value for $z$.

Tips and Tricks

To solve for $z$ efficiently, follow these tips and tricks:

• Isolate the square root term as soon as possible.
• Simplify the equation by combining like terms.
• Square both sides of the equation to solve for $z$.
• Check your work by plugging the solution back into the original equation.

By following these tips and tricks, you'll be well on your way to becoming a master of solving for $z$.

Conclusion

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Introduction

In our previous article, we explored the concept of solving for $z$ using a simple yet effective approach. We broke down the equation $3 \sqrt{z} - 8 = \sqrt{z}$ into smaller, more manageable parts and solved for $z$. In this article, we will delve into a Q&A format, addressing common questions and concerns that readers may have.

Q: What is the value of $z$?

A: The value of $z$ is $16$. This is the solution to the equation $3 \sqrt{z} - 8 = \sqrt{z}$.

Q: How do I isolate the square root term?

A: To isolate the square root term, add $8$ to both sides of the equation. This will simplify the equation and allow you to focus on the square root term.

Q: What is the difference between $2 \sqrt{z}$ and $2z$?

A: $2 \sqrt{z}$ represents the square root of $z$ multiplied by $2$, while $2z$ represents $z$ multiplied by $2$. In this case, we are dealing with the square root term, so we need to be careful not to confuse the two.

Q: Why do I need to square both sides of the equation?

A: Squaring both sides of the equation is necessary to solve for $z$. By squaring both sides, we eliminate the square root term and are left with a simple equation that we can solve.

Q: What are some common mistakes to avoid when solving for $z$?

A: Some common mistakes to avoid when solving for $z$ include:

• Forgetting to isolate the square root term
• Neglecting to square both sides of the equation
• Confusing the square root term with the variable $z$

Q: How can I apply the concept of solving for $z$ to real-world problems?

A: The concept of solving for $z$ has numerous real-world applications. In physics, for example, the value of $z$ can represent the distance between two objects. In engineering, it can represent the length of a wire or cable. By understanding how to solve for $z$, you can apply this knowledge to a wide range of problems and scenarios.

Q: What are some tips and tricks for solving for $z$?

A: Some tips and tricks for solving for $z$ include:

• Isolating the square root term as soon as possible
• Simplifying the equation by combining like terms
• Squaring both sides of the equation to solve for $z$
• Checking your work by plugging the solution back into the original equation

Q: How can I practice solving for $z$?

A: To practice solving for $z$, try working through a series of equations that involve the square root term. Start with simple equations and gradually move on to more complex ones. You can also try using online resources or math software to generate practice problems.

Conclusion

In conclusion, solving for $z$ is a straightforward process that requires patience, persistence, and practice. By following the steps outlined in this article and avoiding common mistakes, you'll be able to solve for $z$ with ease and apply this knowledge to a wide range of problems and scenarios. Remember to practice regularly and seek help when needed to become a master of algebra.

Additional Resources

For further practice and review, try the following resources:

• Khan Academy: Algebra
• Mathway: Algebra Solver
• Wolfram Alpha: Algebra Calculator

By following these resources and practicing regularly, you'll be well on your way to becoming a master of algebra and solving for $z$ with confidence.