(Simplify): ( X 2 Y 2 X 3 Y ) 3 \left(\frac{x^2 Y^2}{x^3 Y}\right)^3 ( X 3 Y X 2 Y 2 ​ ) 3

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements more efficiently. One common technique for simplifying expressions is to apply the rules of exponents, which govern how we handle powers of numbers and variables. In this article, we will focus on simplifying the algebraic expression $\left(\frac{x^2 y^2}{x^3 y}\right)^3$ using the rules of exponents.

Understanding the Rules of Exponents

Before we dive into simplifying the given expression, let's review the basic rules of exponents. The rules of exponents state that when we multiply two powers with the same base, we add their exponents. Conversely, when we divide two powers with the same base, we subtract their exponents. Additionally, when we raise a power to another power, we multiply their exponents.

Rule 1: Multiplying Powers with the Same Base

When we multiply two powers with the same base, we add their exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$.

Rule 2: Dividing Powers with the Same Base

When we divide two powers with the same base, we subtract their exponents. For example, $\frac{x^2}{x^3} = x^{2-3} = x^{-1}$.

Rule 3: Raising a Power to Another Power

When we raise a power to another power, we multiply their exponents. For example, $(x^2)^3 = x^{2\cdot3} = x^6$.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's apply them to simplify the given expression $\left(\frac{x^2 y^2}{x^3 y}\right)^3$. To simplify this expression, we will first apply the rule of dividing powers with the same base to the numerator and denominator.

Step 1: Simplify the Numerator and Denominator

Using the rule of dividing powers with the same base, we can simplify the numerator and denominator as follows:

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

Step 2: Raise the Simplified Expression to the Power of 3

Now that we have simplified the numerator and denominator, we can raise the resulting expression to the power of 3 using the rule of raising a power to another power.

$$\left(\frac{y}{x}\right)^3 = \left(y^1\right)^3 \cdot \left(x^{-1}\right)^3 = y^{3\cdot1} \cdot x^{-3\cdot1} = y^3 \cdot x^{-3} = \frac{y^3}{x^3}$$

Conclusion

In this article, we simplified the algebraic expression $\left(\frac{x^2 y^2}{x^3 y}\right)^3$ using the rules of exponents. We first applied the rule of dividing powers with the same base to simplify the numerator and denominator, and then raised the resulting expression to the power of 3 using the rule of raising a power to another power. The simplified expression is $\frac{y^3}{x^3}$.

Final Answer

The final answer is $\boxed{\frac{y^3}{x^3}}$.

Introduction

In our previous article, we simplified the algebraic expression $\left(\frac{x^2 y^2}{x^3 y}\right)^3$ using the rules of exponents. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the simplification process.

Q&A

Q1: What is the rule of exponents, and how is it used in simplifying expressions?

A1: The rule of exponents states that when we multiply two powers with the same base, we add their exponents. Conversely, when we divide two powers with the same base, we subtract their exponents. Additionally, when we raise a power to another power, we multiply their exponents. These rules are essential in simplifying expressions and solving equations.

Q2: How do we simplify the expression $\left(\frac{x^2 y^2}{x^3 y}\right)^3$ using the rules of exponents?

A2: To simplify the expression, we first apply the rule of dividing powers with the same base to the numerator and denominator. This gives us $\frac{x^2 y^2}{x^3 y} = x^{2-3} \cdot y^{2-1} = x^{-1} \cdot y^1 = \frac{y}{x}$. Then, we raise the resulting expression to the power of 3 using the rule of raising a power to another power.

Q3: What is the difference between multiplying and dividing powers with the same base?

A3: When we multiply two powers with the same base, we add their exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$. Conversely, when we divide two powers with the same base, we subtract their exponents. For example, $\frac{x^2}{x^3} = x^{2-3} = x^{-1}$.

Q4: How do we handle negative exponents in expressions?

A4: Negative exponents can be handled by taking the reciprocal of the base. For example, $x^{-1} = \frac{1}{x}$.

Q5: Can we simplify expressions with variables in the denominator?

A5: Yes, we can simplify expressions with variables in the denominator by applying the rules of exponents. For example, $\frac{x^2}{x^3} = x^{2-3} = x^{-1} = \frac{1}{x}$.

Additional Examples

Example 1: Simplify the expression $\left(\frac{x^3 y^2}{x^2 y^3}\right)^2$

To simplify this expression, we first apply the rule of dividing powers with the same base to the numerator and denominator. This gives us $\frac{x^3 y^2}{x^2 y^3} = x^{3-2} \cdot y^{2-3} = x^1 \cdot y^{-1} = \frac{x}{y}$. Then, we raise the resulting expression to the power of 2 using the rule of raising a power to another power.

Example 2: Simplify the expression $\left(\frac{x^4 y^3}{x^2 y^4}\right)^3$

To simplify this expression, we first apply the rule of dividing powers with the same base to the numerator and denominator. This gives us $\frac{x^4 y^3}{x^2 y^4} = x^{4-2} \cdot y^{3-4} = x^2 \cdot y^{-1} = \frac{x^2}{y}$. Then, we raise the resulting expression to the power of 3 using the rule of raising a power to another power.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional insights into the simplification process of the algebraic expression $\left(\frac{x^2 y^2}{x^3 y}\right)^3$. We also provided additional examples to demonstrate the application of the rules of exponents in simplifying expressions.

Final Answer

The final answer is $\boxed{\frac{y^3}{x^3}}$.