Find All Real Number Solutions For The Equation:$\[ X^{\frac{2}{3}} - X^{\frac{1}{3}} - 6 = 0 \\]If There Is More Than One Solution, Separate Them With Commas.$\[ X = \\]

Introduction

In this article, we will delve into finding real number solutions for the given equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$. This equation involves fractional exponents, which can be challenging to solve. We will use various mathematical techniques to simplify the equation and find its real number solutions.

Understanding the Equation

The given equation is $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$. To begin solving this equation, we need to understand the properties of fractional exponents. A fractional exponent $a^{\frac{m}{n}}$ can be rewritten as $(a^{\frac{1}{n}})^m$. Using this property, we can rewrite the given equation as $(x^{\frac{1}{3}})^2 - x^{\frac{1}{3}} - 6 = 0$.

Simplifying the Equation

Let's simplify the equation by introducing a new variable $y = x^{\frac{1}{3}}$. Substituting this into the equation, we get $y^2 - y - 6 = 0$. This is a quadratic equation in terms of $y$, and we can solve it using the quadratic formula or factoring.

Factoring the Quadratic Equation

The quadratic equation $y^2 - y - 6 = 0$ can be factored as $(y - 3)(y + 2) = 0$. This gives us two possible solutions for $y$: $y = 3$ and $y = -2$.

Finding the Real Number Solutions

Now that we have found the solutions for $y$, we can substitute back to find the real number solutions for $x$. Recall that $y = x^{\frac{1}{3}}$. Therefore, we have $x^{\frac{1}{3}} = 3$ and $x^{\frac{1}{3}} = -2$.

Solving for $x$

To solve for $x$, we need to cube both sides of the equations. For the first equation, we have $x = 3^3 = 27$. For the second equation, we have $x = (-2)^3 = -8$.

Conclusion

In this article, we have found the real number solutions for the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$. The solutions are $x = 27$ and $x = -8$. These solutions were obtained by simplifying the equation using fractional exponents, introducing a new variable, factoring the quadratic equation, and solving for $x$.

Real Number Solutions

The real number solutions for the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ are:

• $x = 27$
• $x = -8$

Graphical Representation

The graph of the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ can be represented as follows:

• The graph has two x-intercepts at $x = 27$ and $x = -8$.
• The graph is a curve that opens upwards.

Applications

The equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ has various applications in mathematics and other fields. For example:

• In algebra, the equation can be used to demonstrate the use of fractional exponents and quadratic equations.
• In calculus, the equation can be used to find the derivative of a function involving fractional exponents.
• In physics, the equation can be used to model real-world phenomena involving fractional exponents.

Conclusion

In conclusion, the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ has two real number solutions: $x = 27$ and $x = -8$. These solutions were obtained by simplifying the equation using fractional exponents, introducing a new variable, factoring the quadratic equation, and solving for $x$. The equation has various applications in mathematics and other fields, and its graphical representation can be used to visualize the solutions.

Q: What is the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$?

A: The equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ is a cubic equation that involves fractional exponents. It can be rewritten as $(x^{\frac{1}{3}})^2 - x^{\frac{1}{3}} - 6 = 0$.

Q: How do I solve the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$?

A: To solve the equation, we can introduce a new variable $y = x^{\frac{1}{3}}$. Substituting this into the equation, we get $y^2 - y - 6 = 0$. This is a quadratic equation in terms of $y$, and we can solve it using the quadratic formula or factoring.

Q: What are the real number solutions for the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$?

A: The real number solutions for the equation are $x = 27$ and $x = -8$.

Q: How do I find the real number solutions for the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$?

A: To find the real number solutions, we need to cube both sides of the equations $y = 3$ and $y = -2$. This gives us $x = 3^3 = 27$ and $x = (-2)^3 = -8$.

Q: What is the significance of the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$?

A: The equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ has various applications in mathematics and other fields. For example, it can be used to demonstrate the use of fractional exponents and quadratic equations, and to find the derivative of a function involving fractional exponents.

Q: Can I use the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ to model real-world phenomena?

A: Yes, the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ can be used to model real-world phenomena involving fractional exponents. For example, it can be used to model the growth of a population that is affected by a fractional exponent.

Q: How do I graph the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$?

A: To graph the equation, we can use a graphing calculator or software. The graph of the equation has two x-intercepts at $x = 27$ and $x = -8$, and it is a curve that opens upwards.

Q: Can I use the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ to solve other equations involving fractional exponents?

A: Yes, the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ can be used to solve other equations involving fractional exponents. For example, it can be used to solve the equation $x^{\frac{3}{2}} - x^{\frac{1}{2}} - 9 = 0$.

Q: How do I simplify the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$?

A: To simplify the equation, we can introduce a new variable $y = x^{\frac{1}{3}}$. Substituting this into the equation, we get $y^2 - y - 6 = 0$. This is a quadratic equation in terms of $y$, and we can solve it using the quadratic formula or factoring.

Q: Can I use the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ to find the derivative of a function involving fractional exponents?

A: Yes, the equation $x^{\frac{2}{3}} - x^{\frac{1}{3}} - 6 = 0$ can be used to find the derivative of a function involving fractional exponents. For example, it can be used to find the derivative of the function $f(x) = x^{\frac{3}{2}}$.