Find The Positive Solution Of The Equation:${ 8x^{\frac{8}{9}} + 9 = 524297 }$

Introduction

In mathematics, equations often present a challenge to solve, especially when they involve fractional exponents. The given equation, $8x^{\frac{8}{9}} + 9 = 524297$, is a prime example of such a problem. In this article, we will delve into the world of mathematics and explore the steps required to find the positive solution of this equation.

Understanding the Equation

The given equation is a complex one, involving a fractional exponent. To begin solving it, we need to understand the properties of fractional exponents. A fractional exponent, such as $\frac{8}{9}$, can be interpreted as raising a number to a power that is itself a fraction. In this case, we have $x^{\frac{8}{9}}$, which means $x$ raised to the power of $\frac{8}{9}$.

Isolating the Variable

To solve the equation, we need to isolate the variable $x$. The first step is to subtract 9 from both sides of the equation, which gives us:

${ 8x^{\frac{8}{9}} = 524297 - 9 }$

Simplifying the Equation

Now, we simplify the right-hand side of the equation by subtracting 9 from 524297:

${ 8x^{\frac{8}{9}} = 524288 }$

Dividing by 8

To isolate the term with the fractional exponent, we divide both sides of the equation by 8:

${ x^{\frac{8}{9}} = \frac{524288}{8} }$

Simplifying the Right-Hand Side

Now, we simplify the right-hand side of the equation by dividing 524288 by 8:

${ x^{\frac{8}{9}} = 65536 }$

Raising to the Power of 9/8

To eliminate the fractional exponent, we raise both sides of the equation to the power of $\frac{9}{8}$:

${ x = (65536)^{\frac{9}{8}} }$

Simplifying the Right-Hand Side

Now, we simplify the right-hand side of the equation by raising 65536 to the power of $\frac{9}{8}$:

${ x = 65536^{\frac{9}{8}} }$

Calculating the Value

To find the value of $x$, we can use a calculator or a computer program to evaluate the expression $65536^{\frac{9}{8}}$.

Conclusion

In this article, we have explored the steps required to find the positive solution of the equation $8x^{\frac{8}{9}} + 9 = 524297$. By isolating the variable, simplifying the equation, and raising to the power of $\frac{9}{8}$, we have arrived at the solution $x = 65536^{\frac{9}{8}}$. This solution provides a unique insight into the world of mathematics and the power of fractional exponents.

Final Answer

The final answer is: $\boxed{65536^{\frac{9}{8}}}$

Introduction

In our previous article, we explored the steps required to find the positive solution of the equation $8x^{\frac{8}{9}} + 9 = 524297$. In this article, we will address some of the most frequently asked questions related to this equation and provide additional insights into the world of mathematics.

Q: What is the significance of fractional exponents in mathematics?

A: Fractional exponents are a fundamental concept in mathematics, allowing us to raise a number to a power that is itself a fraction. This concept is crucial in solving equations involving fractional exponents, such as the one we explored in our previous article.

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

A: To simplify an equation with a fractional exponent, you need to isolate the term with the fractional exponent and then raise both sides of the equation to the power of the reciprocal of the fractional exponent. In the case of the equation $8x^{\frac{8}{9}} + 9 = 524297$, we isolated the term with the fractional exponent and then raised both sides to the power of $\frac{9}{8}$.

Q: What is the difference between a rational exponent and a fractional exponent?

A: A rational exponent is a number that can be expressed as a fraction, such as $\frac{8}{9}$. A fractional exponent, on the other hand, is a number that is itself a fraction, such as $x^{\frac{8}{9}}$. While the terms are often used interchangeably, a rational exponent is a broader concept that includes fractional exponents.

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

A: To evaluate an expression with a fractional exponent, you need to raise the base number to the power of the fractional exponent. In the case of the expression $65536^{\frac{9}{8}}$, we raised 65536 to the power of $\frac{9}{8}$ to find the value of $x$.

Q: What is the relationship between fractional exponents and logarithms?

A: Fractional exponents and logarithms are closely related concepts in mathematics. In fact, the logarithm of a number can be expressed as a fractional exponent. For example, the logarithm of $x$ to the base $a$ can be expressed as $\log_a x = \frac{\log x}{\log a}$. This relationship is crucial in solving equations involving logarithms and fractional exponents.

Q: Can I use a calculator to evaluate an expression with a fractional exponent?

A: Yes, you can use a calculator to evaluate an expression with a fractional exponent. In fact, most calculators have a built-in function for evaluating expressions with fractional exponents. However, it's essential to ensure that your calculator is set to the correct mode and that you enter the expression correctly.

Q: What are some common applications of fractional exponents in real-world problems?

A: Fractional exponents have numerous applications in real-world problems, including finance, physics, and engineering. For example, in finance, fractional exponents are used to calculate compound interest rates and investment returns. In physics, fractional exponents are used to describe the behavior of particles and systems in quantum mechanics.

Conclusion

In this article, we have addressed some of the most frequently asked questions related to the equation $8x^{\frac{8}{9}} + 9 = 524297$ and provided additional insights into the world of mathematics. We hope that this article has been helpful in clarifying the concepts of fractional exponents and their applications in real-world problems.

Final Answer

The final answer is: $\boxed{65536^{\frac{9}{8}}}$