Simplify The Expression:${ \left(\frac{x^{\frac{3}{2}} Y^0 \cdot X^{\frac{1}{2}} Y 2}{x {\frac{1}{2}} Y {-2}}\right) {\frac{5}{3}} }$

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying a complex expression involving exponents and variables. We will break down the problem into manageable steps, and by the end of this guide, you will be able to simplify the given expression with ease.

Understanding Exponents and Variables

Before we dive into the problem, let's quickly review the basics of exponents and variables. Exponents are a shorthand way of representing repeated multiplication. For example, $x^2$ means $x$ multiplied by itself, or $x \cdot x$. Similarly, $x^3$ means $x$ multiplied by itself three times, or $x \cdot x \cdot x$. Variables, on the other hand, are letters or symbols that represent unknown values. In this problem, we will be working with variables $x$ and $y$.

The Given Expression

The given expression is:

$$\left(\frac{x^{\frac{3}{2}} y^0 \cdot x^{\frac{1}{2}} y^2}{x^{\frac{1}{2}} y^{-2}}\right)^{\frac{5}{3}}$$

This expression involves exponents, variables, and fractions. Our goal is to simplify this expression by combining like terms, canceling out common factors, and applying the rules of exponents.

Step 1: Simplify the Expression Inside the Parentheses

To simplify the expression, we need to start by simplifying the expression inside the parentheses. We can do this by applying the rules of exponents and combining like terms.

$$\frac{x^{\frac{3}{2}} y^0 \cdot x^{\frac{1}{2}} y^2}{x^{\frac{1}{2}} y^{-2}}$$

Using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$, we can combine the exponents of $x$:

$$\frac{x^{\frac{3}{2} + \frac{1}{2}} y^0 \cdot x^{\frac{1}{2}} y^2}{x^{\frac{1}{2}} y^{-2}}$$

Simplifying the exponents, we get:

$$\frac{x^2 \cdot x^{\frac{1}{2}} y^2}{x^{\frac{1}{2}} y^{-2}}$$

Step 2: Apply the Rule of Exponents for Division

When dividing two exponential expressions with the same base, we can subtract the exponents. In this case, we have:

$$\frac{x^2 \cdot x^{\frac{1}{2}} y^2}{x^{\frac{1}{2}} y^{-2}}$$

Using the rule of exponents for division, we can subtract the exponents of $x$:

$$x^{2 - \frac{1}{2}} \cdot y^{2 - (-2)}$$

Simplifying the exponents, we get:

$$x^{\frac{3}{2}} \cdot y^4$$

Step 3: Apply the Power Rule of Exponents

Now that we have simplified the expression inside the parentheses, we can apply the power rule of exponents. The power rule states that $(a^m)^n = a^{m \cdot n}$. In this case, we have:

$$(x^{\frac{3}{2}} \cdot y^4)^{\frac{5}{3}}$$

Using the power rule, we can multiply the exponents:

$$x^{\frac{3}{2} \cdot \frac{5}{3}} \cdot y^{4 \cdot \frac{5}{3}}$$

Simplifying the exponents, we get:

$$x^{\frac{5}{2}} \cdot y^{\frac{20}{3}}$$

Conclusion

In this article, we simplified a complex expression involving exponents and variables. We broke down the problem into manageable steps, applying the rules of exponents and combining like terms. By the end of this guide, you should be able to simplify the given expression with ease. Remember to always start by simplifying the expression inside the parentheses, and then apply the power rule of exponents to simplify the final expression.

Final Answer

The final answer is:

$$x^{\frac{5}{2}} \cdot y^{\frac{20}{3}}$$

This is the simplified expression, and it represents the final answer to the problem.

Frequently Asked Questions

• Q: What is the rule of exponents for division? A: When dividing two exponential expressions with the same base, we can subtract the exponents.
• Q: What is the power rule of exponents? A: The power rule states that $(a^m)^n = a^{m \cdot n}$.
• Q: How do I simplify an expression involving exponents and variables? A: Start by simplifying the expression inside the parentheses, and then apply the power rule of exponents to simplify the final expression.

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Algebraic Manipulation
• Wolfram Alpha: Exponents and Variables

Note: The above article is a rewritten version of the given content, optimized for SEO and readability. The article includes headings, subheadings, and bullet points to make it easier to read and understand. The final answer is provided at the end of the article, along with frequently asked questions and additional resources for further learning.

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In our previous article, we provided a step-by-step guide to simplifying a complex expression involving exponents and variables. In this article, we will answer some of the most frequently asked questions about algebraic manipulation, including simplifying expressions with exponents and variables.

Q&A: Simplifying Expressions with Exponents and Variables

Q: What is the rule of exponents for division?

A: When dividing two exponential expressions with the same base, we can subtract the exponents. For example, if we have $x^m \div x^n$, we can simplify it to $x^{m-n}$.

Q: What is the power rule of exponents?

A: The power rule states that $(a^m)^n = a^{m \cdot n}$. This means that when we raise an exponential expression to a power, we can multiply the exponents. For example, if we have $(x^m)^n$, we can simplify it to $x^{m \cdot n}$.

Q: How do I simplify an expression involving exponents and variables?

A: To simplify an expression involving exponents and variables, start by simplifying the expression inside the parentheses. Then, apply the power rule of exponents to simplify the final expression. Remember to always follow the order of operations (PEMDAS):

1. Parentheses: Simplify the expression inside the parentheses.
2. Exponents: Apply the power rule of exponents.
3. Multiplication and Division: Simplify the expression by multiplying and dividing the terms.
4. Addition and Subtraction: Simplify the expression by adding and subtracting the terms.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents an unknown value, while a constant is a value that does not change. For example, in the expression $x + 5$, $x$ is a variable and $5$ is a constant.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, start by simplifying the expression inside the parentheses. Then, apply the power rule of exponents to simplify the final expression. Remember to always follow the order of operations (PEMDAS).

Q: What is the rule for multiplying exponential expressions with the same base?

A: When multiplying exponential expressions with the same base, we can add the exponents. For example, if we have $x^m \cdot x^n$, we can simplify it to $x^{m+n}$.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, we can rewrite the expression with a positive exponent. For example, if we have $x^{-m}$, we can rewrite it as $\frac{1}{x^m}$.

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Mathway: Algebraic Manipulation
• Wolfram Alpha: Exponents and Variables

Conclusion

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we answered some of the most frequently asked questions about algebraic manipulation, including simplifying expressions with exponents and variables. Remember to always follow the order of operations (PEMDAS) and apply the power rule of exponents to simplify the final expression.

Final Tips

• Practice, practice, practice: The more you practice simplifying expressions, the more comfortable you will become with the rules of exponents and variables.
• Use online resources: There are many online resources available to help you learn and practice algebraic manipulation, including Khan Academy, Mathway, and Wolfram Alpha.
• Seek help when needed: Don't be afraid to ask for help if you are struggling with a particular concept or problem.