Find The Positive Solution Of The Equation. 3 X 5 7 + 5 = 50426 3x^{\frac{5}{7}} + 5 = 50426 3 X 7 5 ​ + 5 = 50426

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Introduction

Solving equations with fractional exponents can be a challenging task, especially when dealing with complex expressions. In this article, we will focus on finding the positive solution of the equation $3x^{\frac{5}{7}} + 5 = 50426$. This equation involves a fractional exponent, which requires a different approach than solving linear or quadratic equations.

Understanding Fractional Exponents

Before diving into the solution, it's essential to understand the concept of fractional exponents. A fractional exponent is a way of expressing a number as a power of another number. For example, $x^{\frac{1}{2}}$ represents the square root of $x$, while $x^{\frac{3}{4}}$ represents the cube root of $x$ raised to the power of 3.

In the given equation, the fractional exponent is $\frac{5}{7}$. This means that the variable $x$ is raised to the power of $\frac{5}{7}$, and the coefficient $3$ is multiplied by the result.

Isolating the Variable

To solve the equation, we need to isolate the variable $x$. The first step is to subtract $5$ from both sides of the equation, which gives us:

$$3x^{\frac{5}{7}} = 50426 - 5$$

$$3x^{\frac{5}{7}} = 50421$$

Eliminating the Coefficient

Next, we need to eliminate the coefficient $3$ by dividing both sides of the equation by $3$. This gives us:

$$x^{\frac{5}{7}} = \frac{50421}{3}$$

$$x^{\frac{5}{7}} = 16807$$

Raising Both Sides to the Power of $\frac{7}{5}$

To eliminate the fractional exponent, we need to raise both sides of the equation to the power of $\frac{7}{5}$. This gives us:

$$x = (16807)^{\frac{7}{5}}$$

Evaluating the Expression

Now, we need to evaluate the expression $(16807)^{\frac{7}{5}}$. To do this, we can use a calculator or a computer program to compute the result.

Using a calculator, we get:

$$(16807)^{\frac{7}{5}} \approx 169$$

Conclusion

In this article, we have found the positive solution of the equation $3x^{\frac{5}{7}} + 5 = 50426$. The solution involves understanding fractional exponents, isolating the variable, eliminating the coefficient, and raising both sides of the equation to the power of $\frac{7}{5}$. The final result is $x \approx 169$.

Tips and Tricks

• When dealing with fractional exponents, it's essential to understand the concept of fractional exponents and how to manipulate them.
• To eliminate the fractional exponent, you can raise both sides of the equation to the power of the reciprocal of the exponent.
• When evaluating expressions with fractional exponents, use a calculator or a computer program to compute the result.

Real-World Applications

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

• Physics: When dealing with complex systems, fractional exponents can be used to model the behavior of the system.
• Engineering: Fractional exponents can be used to design and optimize systems, such as electrical circuits and mechanical systems.
• Finance: Fractional exponents can be used to model the behavior of financial systems, such as stock prices and interest rates.

Final Thoughts

Solving equations with fractional exponents requires a different approach than solving linear or quadratic equations. By understanding the concept of fractional exponents and using the correct techniques, you can solve complex equations and apply the results to real-world problems.

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Introduction

In our previous article, we found the positive solution of the equation $3x^{\frac{5}{7}} + 5 = 50426$. In this article, we will answer some of the most frequently asked questions about solving equations with fractional exponents.

Q: What is a fractional exponent?

A: A fractional exponent is a way of expressing a number as a power of another number. For example, $x^{\frac{1}{2}}$ represents the square root of $x$, while $x^{\frac{3}{4}}$ represents the cube root of $x$ raised to the power of 3.

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

A: To solve an equation with a fractional exponent, you need to isolate the variable, eliminate the coefficient, and raise both sides of the equation to the power of the reciprocal of the exponent.

Q: What is the reciprocal of an exponent?

A: The reciprocal of an exponent is the number that, when multiplied by the original exponent, equals 1. For example, the reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$.

Q: How do I raise both sides of an equation to a power?

A: To raise both sides of an equation to a power, you need to multiply the exponent by the power. For example, if you have the equation $x^{\frac{5}{7}} = 16807$, and you want to raise both sides to the power of $\frac{7}{5}$, you would multiply the exponent by $\frac{7}{5}$, resulting in $x = (16807)^{\frac{7}{5}}$.

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

A: A fractional exponent is a way of expressing a number as a power of another number, while a decimal exponent is a way of expressing a number as a power of another number with a decimal coefficient. For example, $x^{0.5}$ is equivalent to $x^{\frac{1}{2}}$, while $x^{0.3}$ is equivalent to $x^{\frac{3}{10}}$.

Q: Can I use a calculator to solve equations with fractional exponents?

A: Yes, you can use a calculator to solve equations with fractional exponents. However, you need to be careful when entering the equation and the exponent, as the calculator may not display the result correctly.

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

A: Solving equations with fractional exponents has many real-world applications, including physics, engineering, and finance. For example, in physics, fractional exponents can be used to model the behavior of complex systems, while in engineering, fractional exponents can be used to design and optimize systems.

Q: Can I use fractional exponents to solve quadratic equations?

A: Yes, you can use fractional exponents to solve quadratic equations. However, you need to be careful when applying the quadratic formula, as the result may involve fractional exponents.

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

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

• Not isolating the variable correctly
• Not eliminating the coefficient correctly
• Not raising both sides of the equation to the correct power
• Not using the correct reciprocal of the exponent

Conclusion

Solving equations with fractional exponents requires a different approach than solving linear or quadratic equations. By understanding the concept of fractional exponents and using the correct techniques, you can solve complex equations and apply the results to real-world problems.