Solve The Equation.$4(x-2)^{2/3} = 64$The Solution Set Is:

Feb 26, 2025 by ADMIN 59 views

$Solve the Equation: $4(x-2)^{2/3} = 64$$

Introduction

Solving equations involving fractional exponents can be a challenging task, especially when dealing with expressions like $4(x-2)^{2/3} = 64$ . In this article, we will delve into the world of fractional exponents and explore the steps required to solve this equation. We will also discuss the properties of fractional exponents and how they can be used to simplify complex expressions.

Understanding Fractional Exponents

Fractional exponents are a way of expressing roots and powers in a more compact form. For example, the expression $x^{1/2}$ represents the square root of $x$ , while $x^{2/3}$ represents the cube root of $x^2$ . When dealing with fractional exponents, it's essential to understand the properties of exponents and how they can be manipulated.

Properties of Fractional Exponents

There are several properties of fractional exponents that are essential to understand when solving equations involving fractional exponents. These properties include:

Raising a power to a power: When raising a power to a power, we multiply the exponents. For example, $(x^m)^n = x^{mn}$ .
Raising a power to a fractional exponent: When raising a power to a fractional exponent, we take the root of the power. For example, $x^{m/n} = \sqrt[n]{x^m}$ .
Multiplying powers: When multiplying powers, we add the exponents. For example, $x^m \cdot x^n = x^{m+n}$ .

Solving the Equation

Now that we have a good understanding of fractional exponents, let's focus on solving the equation $4(x-2)^{2/3} = 64$ . To solve this equation, we need to isolate the variable $x$ .

Step 1: Divide both sides by 4

The first step in solving the equation is to divide both sides by 4. This will help us isolate the expression $(x-2)^{2/3}$ .

from sympy import symbols, Eq, solve

# Define the variable
x = symbols('x')

# Define the equation
equation = Eq(4*(x-2)**(2/3), 64)

# Divide both sides by 4
equation = Eq((x-2)**(2/3), 16)

Step 2: Raise both sides to the power of 3/2

The next step is to raise both sides to the power of 3/2. This will help us eliminate the fractional exponent.

# Raise both sides to the power of 3/2
equation = Eq((x-2), 16**(3/2))

Step 3: Simplify the right-hand side

The right-hand side of the equation is $16^{3/2}$ . We can simplify this expression by evaluating the exponent.

# Simplify the right-hand side
equation = Eq((x-2), 64)

Step 4: Add 2 to both sides

The final step is to add 2 to both sides of the equation. This will help us isolate the variable $x$ .

# Add 2 to both sides
equation = Eq(x, 66)

Conclusion

Solving equations involving fractional exponents can be a challenging task, but with the right approach and understanding of the properties of fractional exponents, it's possible to solve even the most complex equations. In this article, we have walked through the steps required to solve the equation $4(x-2)^{2/3} = 64$ . We have also discussed the properties of fractional exponents and how they can be used to simplify complex expressions.

Final Answer

The final answer to the equation $4(x-2)^{2/3} = 64$ is $x = 66$ .

Additional Resources

For more information on fractional exponents and how to solve equations involving fractional exponents, check out the following resources:

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

Discussion

What are some common mistakes to avoid when solving equations involving fractional exponents? How can you simplify complex expressions involving fractional exponents? Share your thoughts and experiences in the comments below!

Introduction

Solving equations involving fractional exponents can be a challenging task, but with the right approach and understanding of the properties of fractional exponents, it's possible to solve even the most complex equations. 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 roots and powers in a more compact form. For example, the expression $x^{1/2}$ represents the square root of $x$ , while $x^{2/3}$ represents the cube root of $x^2$ .

Q: How do I simplify a fractional exponent?

A: To simplify a fractional exponent, you need to understand the properties of exponents and how they can be manipulated. For example, when raising a power to a power, we multiply the exponents. When raising a power to a fractional exponent, we take the root of the power.

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

A: A rational exponent is a way of expressing a root and a power in a single expression. For example, the expression $x^{3/2}$ represents the square root of $x^3$ . A fractional exponent is a way of expressing a root and a power in a more compact form.

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. This can be done by using the properties of exponents and manipulating the equation to get rid of the fractional exponent.

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 understanding the properties of exponents and how they can be manipulated
Not isolating the variable
Not checking the solution to make sure it satisfies the original equation

Q: How do I check my solution to an equation with a fractional exponent?

A: To check your solution to an equation with a fractional exponent, you need to plug the solution back into the original equation and make sure it satisfies the equation.

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

A: Fractional exponents have many real-world applications, including:

Physics: Fractional exponents are used to describe the behavior of particles in quantum mechanics.
Engineering: Fractional exponents are used to describe the behavior of systems in control theory.
Finance: Fractional exponents are used to describe the behavior of financial instruments in options pricing.

Q: How do I teach fractional exponents to my students?

A: To teach fractional exponents to your students, you need to start with the basics and build up to more complex concepts. You can use real-world examples and visual aids to help your students understand the concept of fractional exponents.

Q: What are some resources for learning more about fractional exponents?

A: Some resources for learning more about fractional exponents include:

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

Conclusion

Solving equations involving fractional exponents can be a challenging task, but with the right approach and understanding of the properties of fractional exponents, it's possible to solve even the most complex equations. In this article, we have answered some of the most frequently asked questions about solving equations with fractional exponents.

Final Answer

The final answer to the equation $4(x-2)^{2/3} = 64$ is $x = 66$ .

Additional Resources

For more information on fractional exponents and how to solve equations involving fractional exponents, check out the following resources:

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

Discussion

What are some common mistakes to avoid when solving equations with fractional exponents? How can you simplify complex expressions involving fractional exponents? Share your thoughts and experiences in the comments below!