Simplify The Expression:$ X^2 Y^{-4} \cdot X^3 Y^2 $

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Understanding the Problem

When dealing with algebraic expressions, simplifying them is a crucial step in solving mathematical problems. In this article, we will focus on simplifying the given expression: $x^2 y^{-4} \cdot x^3 y^2$. This involves applying the rules of exponents and understanding the properties of negative exponents.

Review of Exponent Rules

Before we dive into simplifying the expression, let's review the rules of exponents. When multiplying two numbers with the same base, we add their exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$. Similarly, when dividing two numbers with the same base, we subtract their exponents. For instance, $\frac{x^2}{x^3} = x^{2-3} = x^{-1}$.

Applying Exponent Rules to the Expression

Now that we have reviewed the rules of exponents, let's apply them to the given expression: $x^2 y^{-4} \cdot x^3 y^2$. We can start by multiplying the coefficients and adding the exponents of the variables with the same base.

Multiplying Coefficients

The coefficients of the expression are $1$ and $1$, which are multiplied together to get $1 \cdot 1 = 1$.

Adding Exponents of Variables with the Same Base

Now, let's focus on the variables. We have $x^2$ and $x^3$, which have the same base ($x$). We can add their exponents to get $x^{2+3} = x^5$.

Similarly, we have $y^{-4}$ and $y^2$, which have the same base ($y$). We can add their exponents to get $y^{-4+2} = y^{-2}$.

Combining the Results

Now that we have simplified the variables, we can combine the results to get the final expression: $1 \cdot x^5 \cdot y^{-2}$.

Understanding Negative Exponents

In the previous step, we obtained the expression $y^{-2}$. This is a negative exponent, which can be rewritten as a positive exponent by taking the reciprocal of the base. In this case, we can rewrite $y^{-2}$ as $\frac{1}{y^2}$.

Simplifying the Expression

Now that we have rewritten the negative exponent, we can simplify the expression further. We have $1 \cdot x^5 \cdot \frac{1}{y^2}$, which can be rewritten as $\frac{x^5}{y^2}$.

Conclusion

In this article, we simplified the expression $x^2 y^{-4} \cdot x^3 y^2$ by applying the rules of exponents and understanding the properties of negative exponents. We obtained the final expression $\frac{x^5}{y^2}$, which is a simplified form of the original expression.

Final Answer

The final answer is $\boxed{\frac{x^5}{y^2}}$.

Frequently Asked Questions

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

A: When multiplying two numbers with the same base, we add their exponents.

Q: What is the rule for dividing numbers with the same base?

A: When dividing two numbers with the same base, we subtract their exponents.

Q: How do we handle negative exponents?

A: We can rewrite negative exponents as positive exponents by taking the reciprocal of the base.

Q: What is the final expression after simplifying $x^2 y^{-4} \cdot x^3 y^2$?

A: The final expression is $\frac{x^5}{y^2}$.

Step-by-Step Solution

1. Multiply the coefficients: $1 \cdot 1 = 1$
2. Add the exponents of the variables with the same base:
   • $x^2$ and $x^3$: $x^{2+3} = x^5$
   • $y^{-4}$ and $y^2$: $y^{-4+2} = y^{-2}$
3. Combine the results: $1 \cdot x^5 \cdot y^{-2}$
4. Rewrite the negative exponent: $y^{-2} = \frac{1}{y^2}$
5. Simplify the expression: $\frac{x^5}{y^2}$
   

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Understanding the Problem

When dealing with algebraic expressions, simplifying them is a crucial step in solving mathematical problems. In this article, we will focus on simplifying the given expression: $x^2 y^{-4} \cdot x^3 y^2$. This involves applying the rules of exponents and understanding the properties of negative exponents.

Q&A Session

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

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

Q: What is the rule for dividing numbers with the same base?

A: When dividing two numbers with the same base, we subtract their exponents. For instance, $\frac{x^2}{x^3} = x^{2-3} = x^{-1}$.

Q: How do we handle negative exponents?

A: We can rewrite negative exponents as positive exponents by taking the reciprocal of the base. For example, $y^{-2} = \frac{1}{y^2}$.

Q: What is the final expression after simplifying $x^2 y^{-4} \cdot x^3 y^2$?

A: The final expression is $\frac{x^5}{y^2}$.

Q: Can we simplify expressions with variables and constants?

A: Yes, we can simplify expressions with variables and constants by applying the rules of exponents. For example, $2x^2 \cdot 3x^3 = 6x^{2+3} = 6x^5$.

Q: How do we handle expressions with multiple variables?

A: We can handle expressions with multiple variables by applying the rules of exponents separately for each variable. For example, $x^2 y^{-4} \cdot x^3 y^2 = x^{2+3} \cdot y^{-4+2} = x^5 y^{-2}$.

Q: Can we simplify expressions with fractional exponents?

A: Yes, we can simplify expressions with fractional exponents by applying the rules of exponents. For example, $x^{\frac{1}{2}} \cdot x^{\frac{3}{2}} = x^{\frac{1}{2} + \frac{3}{2}} = x^2$.

Q: How do we handle expressions with negative coefficients?

A: We can handle expressions with negative coefficients by applying the rules of exponents and multiplying the coefficient by the base raised to the power of the exponent. For example, $-x^2 \cdot x^3 = -x^{2+3} = -x^5$.

Step-by-Step Solution

1. Multiply the coefficients: $1 \cdot 1 = 1$
2. Add the exponents of the variables with the same base:
   • $x^2$ and $x^3$: $x^{2+3} = x^5$
   • $y^{-4}$ and $y^2$: $y^{-4+2} = y^{-2}$
3. Combine the results: $1 \cdot x^5 \cdot y^{-2}$
4. Rewrite the negative exponent: $y^{-2} = \frac{1}{y^2}$
5. Simplify the expression: $\frac{x^5}{y^2}$

Common Mistakes to Avoid

• Not applying the rules of exponents correctly
• Not handling negative exponents properly
• Not simplifying expressions with variables and constants
• Not handling expressions with multiple variables
• Not simplifying expressions with fractional exponents
• Not handling expressions with negative coefficients

Conclusion

In this article, we simplified the expression $x^2 y^{-4} \cdot x^3 y^2$ by applying the rules of exponents and understanding the properties of negative exponents. We also answered some frequently asked questions related to simplifying expressions with variables and constants, multiple variables, fractional exponents, and negative coefficients.

Final Answer

The final answer is $\boxed{\frac{x^5}{y^2}}$.