Find The Real Solution(s) Of \[$(x-2)^5 = 32\$\].

Introduction

Solving equations involving exponents can be challenging, especially when dealing with high-degree polynomials. In this article, we will focus on finding the real solution(s) of the equation {(x-2)^5 = 32$}$. This equation involves a fifth-degree polynomial, and we will use various mathematical techniques to isolate the variable x.

Understanding the Equation

The given equation is {(x-2)^5 = 32$}$. To begin solving this equation, we need to understand the properties of exponents. When a number is raised to a power, it means that the number is multiplied by itself as many times as the power indicates. For example, {x^3 = x \cdot x \cdot x$}$.

In the given equation, the base is {(x-2)$}$, and the exponent is 5. This means that {(x-2)$}$ is multiplied by itself 5 times, resulting in the value 32.

Isolating the Variable

To isolate the variable x, we need to get rid of the exponent. One way to do this is to take the fifth root of both sides of the equation. The fifth root of a number is a value that, when raised to the power of 5, gives the original number.

Taking the fifth root of both sides of the equation, we get:

{\sqrt[5]{(x-2)^5} = \sqrt[5]{32}$}$

Using the property of exponents that states {\sqrt[n]{a^n} = a$}$, we can simplify the left-hand side of the equation:

{x-2 = \sqrt[5]{32}$}$

Simplifying the Right-Hand Side

To simplify the right-hand side of the equation, we need to find the fifth root of 32. The fifth root of 32 is a value that, when raised to the power of 5, gives 32.

{\sqrt[5]{32} = 2$}$

Substituting this value into the equation, we get:

{x-2 = 2$}$

Solving for x

To solve for x, we need to isolate the variable. Adding 2 to both sides of the equation, we get:

{x = 2 + 2$}$

{x = 4$}$

Conclusion

In this article, we have found the real solution(s) of the equation {(x-2)^5 = 32$}$. By using various mathematical techniques, including taking the fifth root of both sides of the equation and simplifying the right-hand side, we have isolated the variable x and found its value.

The final answer is {x = 4$}$. This solution satisfies the original equation, and it is the only real solution.

Additional Tips and Tricks

When solving equations involving exponents, it is essential to understand the properties of exponents and how to manipulate them. Here are some additional tips and tricks to help you solve similar equations:

• Use the property of exponents that states {\sqrt[n]{a^n} = a$}$ to simplify the left-hand side of the equation.
• Use the property of exponents that states {a^m \cdot a^n = a^{m+n}$}$ to combine like terms.
• Use the property of exponents that states {\frac{a^m}{an} = a^{m-n}$}$ to simplify fractions.
• Use the property of exponents that states {(a^m)n = a^{m \cdot n}$}$ to simplify expressions.

By following these tips and tricks, you can solve equations involving exponents with ease.

Frequently Asked Questions

Here are some frequently asked questions about solving equations involving exponents:

• Q: What is the property of exponents that states {\sqrt[n]{a^n} = a$}$?
• A: This property states that the nth root of a number raised to the power of n is equal to the original number.
• Q: How do I simplify the left-hand side of an equation involving exponents?
• A: You can use the property of exponents that states {\sqrt[n]{a^n} = a$}$ to simplify the left-hand side of the equation.
• Q: How do I combine like terms in an equation involving exponents?
• A: You can use the property of exponents that states {a^m \cdot a^n = a^{m+n}$}$ to combine like terms.

By following these tips and tricks, you can solve equations involving exponents with ease.

Final Thoughts

Solving equations involving exponents can be challenging, but with the right techniques and strategies, you can find the real solution(s) of the equation. In this article, we have found the real solution(s) of the equation {(x-2)^5 = 32$}$ by using various mathematical techniques, including taking the fifth root of both sides of the equation and simplifying the right-hand side.

The final answer is {x = 4$}$. This solution satisfies the original equation, and it is the only real solution.

By following the tips and tricks outlined in this article, you can solve equations involving exponents with ease. Remember to use the properties of exponents to simplify the left-hand side of the equation, combine like terms, and simplify fractions. With practice and patience, you can become proficient in solving equations involving exponents.