Find The Value Of $n$ For The Following Equation:$x^5 \sqrt{x} = X^n$\$n =$[/tex\] $\square$

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Introduction

In this article, we will delve into solving for the value of $n$ in the given equation: $x^5 \sqrt{x} = x^n$. This equation involves exponentiation and radicals, making it a challenging problem to solve. We will break down the solution step by step, using algebraic manipulations and properties of exponents to find the value of $n$.

Understanding the Equation

The given equation is $x^5 \sqrt{x} = x^n$. To simplify this equation, we can start by expressing the square root of $x$ as a fractional exponent: $\sqrt{x} = x^{1/2}$. Substituting this into the equation, we get:

$$x^5 x^{1/2} = x^n$$

Applying Exponent Rules

Using the product of powers rule, which states that $a^m a^n = a^{m+n}$, we can combine the exponents on the left-hand side of the equation:

$$x^{5+1/2} = x^n$$

Simplifying the exponent, we get:

$$x^{11/2} = x^n$$

Equating Exponents

Since the bases of the exponents are the same ($x$), we can equate the exponents:

$$\frac{11}{2} = n$$

Conclusion

Therefore, the value of $n$ in the equation $x^5 \sqrt{x} = x^n$ is $\boxed{\frac{11}{2}}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Express the square root of $x$ as a fractional exponent: $\sqrt{x} = x^{1/2}$
2. Substitute the fractional exponent into the equation: $x^5 x^{1/2} = x^n$
3. Apply the product of powers rule: $x^{5+1/2} = x^n$
4. Simplify the exponent: $x^{11/2} = x^n$
5. Equating exponents: $\frac{11}{2} = n$

Tips and Tricks

• When dealing with equations involving exponents and radicals, it's essential to simplify the expressions using exponent rules and properties of radicals.
• Equating exponents is a powerful technique for solving equations involving the same base.
• Practice solving equations involving exponents and radicals to become more comfortable with these types of problems.

Real-World Applications

Solving equations involving exponents and radicals has numerous real-world applications in fields such as:

• Physics: Exponents and radicals are used to describe physical phenomena, such as the motion of objects and the behavior of electrical circuits.
• Engineering: Exponents and radicals are used to design and analyze complex systems, such as bridges and buildings.
• Computer Science: Exponents and radicals are used in algorithms and data structures to solve problems efficiently.

Conclusion

In conclusion, solving for $n$ in the equation $x^5 \sqrt{x} = x^n$ requires careful manipulation of exponents and radicals. By applying exponent rules and equating exponents, we can find the value of $n$ as $\boxed{\frac{11}{2}}$. This problem demonstrates the importance of understanding exponent rules and properties of radicals in solving equations involving exponents and radicals.

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Q: What is the main concept behind solving for n in the equation $x^5 \sqrt{x} = x^n$?

A: The main concept behind solving for n in the equation $x^5 \sqrt{x} = x^n$ is to simplify the expression using exponent rules and properties of radicals, and then equate the exponents to find the value of n.

Q: How do I simplify the expression $x^5 \sqrt{x}$ using exponent rules?

A: To simplify the expression $x^5 \sqrt{x}$, you can express the square root of x as a fractional exponent: $\sqrt{x} = x^{1/2}$. Then, you can apply the product of powers rule to combine the exponents: $x^5 x^{1/2} = x^{5+1/2}$.

Q: What is the product of powers rule, and how is it used in solving for n?

A: The product of powers rule states that $a^m a^n = a^{m+n}$. In solving for n, we use this rule to combine the exponents on the left-hand side of the equation: $x^5 x^{1/2} = x^{5+1/2}$.

Q: How do I equate the exponents to find the value of n?

A: To equate the exponents, we set the exponents on both sides of the equation equal to each other: $\frac{11}{2} = n$.

Q: What are some common mistakes to avoid when solving for n in the equation $x^5 \sqrt{x} = x^n$?

A: Some common mistakes to avoid when solving for n in the equation $x^5 \sqrt{x} = x^n$ include:

• Not simplifying the expression using exponent rules and properties of radicals
• Not equating the exponents correctly
• Not checking the validity of the solution

Q: How can I practice solving equations involving exponents and radicals?

A: You can practice solving equations involving exponents and radicals by:

• Working through example problems
• Using online resources and practice exercises
• Solving real-world problems that involve exponents and radicals

Q: What are some real-world applications of solving equations involving exponents and radicals?

A: Some real-world applications of solving equations involving exponents and radicals include:

• Physics: Exponents and radicals are used to describe physical phenomena, such as the motion of objects and the behavior of electrical circuits.
• Engineering: Exponents and radicals are used to design and analyze complex systems, such as bridges and buildings.
• Computer Science: Exponents and radicals are used in algorithms and data structures to solve problems efficiently.

Q: Can I use a calculator to solve for n in the equation $x^5 \sqrt{x} = x^n$?

A: While a calculator can be used to simplify the expression and calculate the value of n, it is not necessary to use a calculator to solve for n. By applying exponent rules and equating exponents, you can find the value of n without using a calculator.

Q: How can I check the validity of the solution for n?

A: To check the validity of the solution for n, you can:

• Plug the value of n back into the original equation to see if it is true
• Check if the solution satisfies the original equation
• Use a calculator to verify the solution

Q: What are some additional tips for solving equations involving exponents and radicals?

A: Some additional tips for solving equations involving exponents and radicals include:

• Simplifying the expression using exponent rules and properties of radicals
• Equating the exponents correctly
• Checking the validity of the solution
• Practicing solving equations involving exponents and radicals to become more comfortable with these types of problems.