Solve For \[$ X \$\]:$\[ X^{2 / 3} = \frac{9}{4} \\]

Introduction

Solving for x in an equation involving fractional exponents can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will explore how to solve for x in the equation $x^{2/3} = \frac{9}{4}$, which involves a fractional exponent. We will use algebraic manipulations and properties of exponents to isolate x and find its value.

Understanding Fractional Exponents

Before we dive into solving the equation, let's take a moment to understand what fractional exponents represent. A fractional exponent, such as $a^{m/n}$, can be interpreted as taking the nth root of a and raising it to the power of m. In other words, $a^{m/n} = (a^{1/n})^m$. This property will be useful in solving the equation.

Solving the Equation

To solve the equation $x^{2/3} = \frac{9}{4}$, we can start by isolating the variable x. We can do this by raising both sides of the equation to the power of 3/2, which is the reciprocal of the fractional exponent. This will eliminate the fractional exponent and allow us to solve for x.

x^{2/3} = \frac{9}{4}

Raising both sides to the power of 3/2:

(x^{2/3})^{3/2} = (\frac{9}{4})^{3/2}

Using the property of fractional exponents, we can simplify the left-hand side:

x^1 = (\frac{9}{4})^{3/2}

Now, we can simplify the right-hand side by evaluating the expression:

x = (\frac{9}{4})^{3/2}

To evaluate this expression, we can use the property of exponents that states $(a/b)^n = (a^n)/(b^n)$. Applying this property, we get:

x = \frac{(9)^{3/2}}{(4)^{3/2}}

Simplifying the numerator and denominator, we get:

x = \frac{27}{8}

Conclusion

In this article, we solved for x in the equation $x^{2/3} = \frac{9}{4}$ by using algebraic manipulations and properties of exponents. We raised both sides of the equation to the power of 3/2, eliminated the fractional exponent, and simplified the expression to find the value of x. The final answer is $\boxed{\frac{27}{8}}$.

Additional Tips and Tricks

• When solving equations involving fractional exponents, it's essential to remember the property of fractional exponents, which states that $a^{m/n} = (a^{1/n})^m$.
• To eliminate a fractional exponent, raise both sides of the equation to the power of the reciprocal of the fractional exponent.
• When simplifying expressions involving exponents, use the property of exponents that states $(a/b)^n = (a^n)/(b^n)$.

Frequently Asked Questions

• Q: What is the value of x in the equation $x^{2/3} = \frac{9}{4}$? A: The value of x is $\boxed{\frac{27}{8}}$.
• Q: How do I eliminate a fractional exponent in an equation? A: To eliminate a fractional exponent, raise both sides of the equation to the power of the reciprocal of the fractional exponent.
• Q: What is the property of fractional exponents? A: The property of fractional exponents states that $a^{m/n} = (a^{1/n})^m$.
  

Introduction

In our previous article, we solved for x in the equation $x^{2/3} = \frac{9}{4}$ using algebraic manipulations and properties of exponents. However, we know that math can be a challenging subject, and sometimes, it's not enough to just provide a solution. That's why we've put together this Q&A guide to help you better understand the concepts and techniques involved in solving equations with fractional exponents.

Q&A: Solving Equations with Fractional Exponents

Q: What is a fractional exponent, and how is it different from a regular exponent?

A: A fractional exponent is an exponent that is a fraction, such as $a^{m/n}$. It can be interpreted as taking the nth root of a and raising it to the power of m. For example, $a^{2/3}$ can be written as $(a^{1/3})^2$.

Q: How do I eliminate a fractional exponent in an equation?

A: To eliminate a fractional exponent, raise both sides of the equation to the power of the reciprocal of the fractional exponent. For example, to eliminate the fractional exponent in the equation $x^{2/3} = \frac{9}{4}$, we can raise both sides to the power of 3/2.

Q: What is the property of fractional exponents, and how can I use it to simplify expressions?

A: The property of fractional exponents states that $a^{m/n} = (a^{1/n})^m$. This means that we can rewrite a fractional exponent as a product of two exponents: one that represents the nth root of a, and another that represents the power of m. For example, $a^{2/3}$ can be rewritten as $(a^{1/3})^2$.

Q: How do I simplify expressions involving exponents?

A: To simplify expressions involving exponents, use the property of exponents that states $(a/b)^n = (a^n)/(b^n)$. This means that we can rewrite a fraction with an exponent as a product of two fractions: one that represents the numerator raised to the power of n, and another that represents the denominator raised to the power of n.

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 raising both sides of the equation to the power of the reciprocal of the fractional exponent
• Not using the property of fractional exponents to simplify expressions
• Not checking the domain of the expression to ensure that it is defined for all values of x

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

A: To practice solving equations with fractional exponents, try working through some sample problems, such as:

• $x^{2/3} = \frac{9}{4}$
• $x^{3/4} = \frac{27}{8}$
• $x^{1/2} = \frac{9}{4}$

You can also try using online resources, such as math websites or video tutorials, to get additional practice and support.

Conclusion

Solving equations with fractional exponents can be a challenging task, but with the right approach and practice, it can be mastered. By understanding the property of fractional exponents and using algebraic manipulations, we can simplify expressions and solve equations with fractional exponents. We hope that this Q&A guide has been helpful in providing you with a better understanding of the concepts and techniques involved in solving equations with fractional exponents.

Additional Resources

• Khan Academy: Solving Equations with Fractional Exponents
• Mathway: Solving Equations with Fractional Exponents
• Wolfram Alpha: Solving Equations with Fractional Exponents

Frequently Asked Questions

• Q: What is the value of x in the equation $x^{2/3} = \frac{9}{4}$? A: The value of x is $\boxed{\frac{27}{8}}$.
• Q: How do I eliminate a fractional exponent in an equation? A: To eliminate a fractional exponent, raise both sides of the equation to the power of the reciprocal of the fractional exponent.
• Q: What is the property of fractional exponents? A: The property of fractional exponents states that $a^{m/n} = (a^{1/n})^m$.