Solve The Equation: $\[ 334 = K^{\frac{3}{2}} - 9 \\]

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Introduction

Solving equations involving fractional exponents can be a challenging task, especially when dealing with complex expressions. In this article, we will focus on solving the equation $334 = k^{\frac{3}{2}} - 9$, where $k$ is the variable we need to isolate. We will use algebraic manipulations and properties of exponents to simplify the equation and find the value of $k$.

Understanding the Equation

The given equation is $334 = k^{\frac{3}{2}} - 9$. To start solving this equation, we need to isolate the variable $k$. The first step is to add $9$ to both sides of the equation, which gives us:

$$334 + 9 = k^{\frac{3}{2}}$$

This simplifies to:

$$343 = k^{\frac{3}{2}}$$

Using Properties of Exponents

Now that we have isolated the variable $k$ on one side of the equation, we can use properties of exponents to simplify the expression. Specifically, we can use the property that states $a^m \cdot a^n = a^{m+n}$.

In this case, we can rewrite the equation as:

$$k^{\frac{3}{2}} = 343$$

Since $343$ is equal to $7^3$, we can rewrite the equation as:

$$k^{\frac{3}{2}} = 7^3$$

Simplifying the Expression

Now that we have rewritten the equation in terms of powers of $7$, we can use the property of exponents that states $(a^m)^n = a^{mn}$.

In this case, we can rewrite the equation as:

$$(k^{\frac{1}{2}})^3 = 7^3$$

This simplifies to:

$$k^{\frac{1}{2}} = 7$$

Isolating the Variable

Now that we have simplified the expression, we can isolate the variable $k$ by squaring both sides of the equation. This gives us:

$$(k^{\frac{1}{2}})^2 = 7^2$$

This simplifies to:

$$k = 49$$

Conclusion

In this article, we have solved the equation $334 = k^{\frac{3}{2}} - 9$ using algebraic manipulations and properties of exponents. We have isolated the variable $k$ and found its value to be $49$. This demonstrates the importance of understanding properties of exponents and how to apply them to solve complex equations.

Additional Tips and Tricks

• When dealing with equations involving fractional exponents, it is often helpful to use the property that states $a^m \cdot a^n = a^{m+n}$.
• When simplifying expressions involving powers of $a$, it is often helpful to use the property that states $(a^m)^n = a^{mn}$.
• When isolating variables, it is often helpful to use inverse operations, such as squaring both sides of the equation.

Real-World Applications

Solving equations involving fractional exponents has many real-world applications, including:

• Physics: When dealing with problems involving motion, energy, and momentum, fractional exponents are often used to describe the relationships between different physical quantities.
• Engineering: When designing systems, engineers often use fractional exponents to describe the relationships between different components and their performance.
• Computer Science: When writing algorithms, programmers often use fractional exponents to describe the relationships between different data structures and their performance.

Final Thoughts

Solving equations involving fractional exponents can be a challenging task, but with practice and patience, it can be mastered. By understanding properties of exponents and applying them to solve complex equations, we can gain a deeper understanding of the underlying mathematics and develop problem-solving skills that can be applied to a wide range of real-world problems.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Introduction to Mathematical Thinking" by Keith Devlin

Further Reading

• For more information on solving equations involving fractional exponents, see the following resources:

• Khan Academy: Solving Equations with Fractional Exponents
• MIT OpenCourseWare: Solving Equations with Fractional Exponents
• Wolfram MathWorld: Solving Equations with Fractional Exponents
  

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Introduction

In our previous article, we solved the equation $334 = k^{\frac{3}{2}} - 9$ using algebraic manipulations and properties of exponents. In this article, we will answer some of the most frequently asked questions about solving equations involving fractional exponents.

Q: What is a fractional exponent?

A: A fractional exponent is an exponent that is a fraction, such as $\frac{1}{2}$ or $\frac{3}{4}$. Fractional exponents are used to describe the relationships between different physical quantities, such as distance, time, and velocity.

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, you can use the property that states $a^m \cdot a^n = a^{m+n}$. For example, if you have the expression $x^{\frac{1}{2}} \cdot x^{\frac{1}{2}}$, you can simplify it by adding the exponents: $x^{\frac{1}{2} + \frac{1}{2}} = x^1 = x$.

Q: How do I isolate a variable with a fractional exponent?

A: To isolate a variable with a fractional exponent, you can use inverse operations, such as squaring both sides of the equation. For example, if you have the equation $x^{\frac{1}{2}} = 4$, you can isolate the variable by squaring both sides: $(x^{\frac{1}{2}})^2 = 4^2$, which simplifies to $x = 16$.

Q: What are some common mistakes to avoid when solving equations involving fractional exponents?

A: Some common mistakes to avoid when solving equations involving fractional exponents include:

• Not using the correct property of exponents, such as $a^m \cdot a^n = a^{m+n}$.
• Not simplifying the expression correctly, such as not adding the exponents.
• Not using inverse operations, such as not squaring both sides of the equation.

Q: How do I apply the properties of exponents to solve real-world problems?

A: To apply the properties of exponents to solve real-world problems, you can use the following steps:

1. Identify the relationships between the different physical quantities in the problem.
2. Use the properties of exponents to describe these relationships.
3. Simplify the expression using the properties of exponents.
4. Use inverse operations to isolate the variable.

Q: What are some real-world applications of solving equations involving fractional exponents?

A: Some real-world applications of solving equations involving fractional exponents include:

• Physics: Solving equations involving fractional exponents is used to describe the relationships between different physical quantities, such as distance, time, and velocity.
• Engineering: Solving equations involving fractional exponents is used to design systems and predict their performance.
• Computer Science: Solving equations involving fractional exponents is used to write algorithms and predict their performance.

Q: How can I practice solving equations involving fractional exponents?

A: To practice solving equations involving fractional exponents, you can try the following:

• Work through practice problems in a textbook or online resource.
• Use online tools, such as calculators or graphing software, to visualize the relationships between different physical quantities.
• Try solving real-world problems, such as designing a system or predicting the performance of an algorithm.

Conclusion

Solving equations involving fractional exponents can be a challenging task, but with practice and patience, it can be mastered. By understanding properties of exponents and applying them to solve complex equations, we can gain a deeper understanding of the underlying mathematics and develop problem-solving skills that can be applied to a wide range of real-world problems.

Additional Resources

• For more information on solving equations involving fractional exponents, see the following resources:

• Khan Academy: Solving Equations with Fractional Exponents
• MIT OpenCourseWare: Solving Equations with Fractional Exponents
• Wolfram MathWorld: Solving Equations with Fractional Exponents

• For practice problems and exercises, see the following resources:

• Mathway: Solving Equations with Fractional Exponents
• Symbolab: Solving Equations with Fractional Exponents
• IXL: Solving Equations with Fractional Exponents