Find The Product: $\left(x^9 Y^{-15}\right)\left(x^{-12} Y^{10}\right) =$

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Understanding the Problem

When dealing with exponents, we need to apply the rules of multiplication to simplify the expression. The given problem involves multiplying two expressions with exponents, and we need to find the product.

Rules of Exponent Multiplication

To multiply two expressions with exponents, we need to follow the rules of exponent multiplication. The rules state that when multiplying two expressions with the same base, we add the exponents. However, when the bases are different, we multiply the expressions as they are.

Applying the Rules of Exponent Multiplication

In this problem, we have two expressions with exponents: $\left(x^9 y^{-15}\right)$ and $\left(x^{-12} y^{10}\right)$. We can start by multiplying the expressions with the same base, which is $x$. We add the exponents $9$ and $-12$ to get $-3$. So, the product of the expressions with the same base is $x^{-3}$.

Multiplying the Expressions with Different Bases

Now, we need to multiply the expressions with different bases, which are $y$. We multiply the expressions as they are, since the bases are different. The product of the expressions with different bases is $y^{-15} \cdot y^{10} = y^{-5}$.

Combining the Results

Now, we can combine the results of multiplying the expressions with the same base and the expressions with different bases. The product of the two expressions is $x^{-3} \cdot y^{-5}$.

Simplifying the Expression

We can simplify the expression by combining the exponents. Since the bases are the same, we can add the exponents. The product of the two expressions is $x^{-3} \cdot y^{-5} = x^{-8} y^{-5}$.

Conclusion

In conclusion, the product of the two expressions $\left(x^9 y^{-15}\right)\left(x^{-12} y^{10}\right)$ is $x^{-8} y^{-5}$.

Example Use Case

This problem can be used to demonstrate the rules of exponent multiplication. It can also be used to help students understand how to simplify expressions with exponents.

Tips and Tricks

• When multiplying two expressions with exponents, make sure to follow the rules of exponent multiplication.
• When the bases are the same, add the exponents.
• When the bases are different, multiply the expressions as they are.
• Simplify the expression by combining the exponents.

Common Mistakes

• Failing to follow the rules of exponent multiplication.
• Not adding the exponents when the bases are the same.
• Not multiplying the expressions as they are when the bases are different.

Real-World Applications

This problem can be used in real-world applications such as:

• Simplifying expressions in algebra and calculus.
• Solving equations and inequalities.
• Working with exponents in finance and economics.

Final Answer

The final answer is: $\boxed{x^{-8} y^{-5}}$

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Frequently Asked Questions

Q: What are the rules of exponent multiplication?

A: The rules of exponent multiplication state that when multiplying two expressions with the same base, we add the exponents. However, when the bases are different, we multiply the expressions as they are.

Q: How do I multiply two expressions with exponents?

A: To multiply two expressions with exponents, you need to follow the rules of exponent multiplication. If the bases are the same, add the exponents. If the bases are different, multiply the expressions as they are.

Q: What is the product of $\left(x^9 y^{-15}\right)\left(x^{-12} y^{10}\right)$?

A: The product of $\left(x^9 y^{-15}\right)\left(x^{-12} y^{10}\right)$ is $x^{-8} y^{-5}$.

Q: Why do I need to add the exponents when the bases are the same?

A: When the bases are the same, we add the exponents because it represents the same quantity being multiplied together. For example, $x^9 \cdot x^{-12} = x^{9-12} = x^{-3}$.

Q: Why do I need to multiply the expressions as they are when the bases are different?

A: When the bases are different, we multiply the expressions as they are because it represents different quantities being multiplied together. For example, $x^9 \cdot y^{-15} = x^9 \cdot y^{-15}$.

Q: Can I simplify the expression $x^{-8} y^{-5}$?

A: Yes, you can simplify the expression $x^{-8} y^{-5}$ by combining the exponents. Since the bases are the same, you can add the exponents. The simplified expression is $x^{-8} y^{-5} = \frac{1}{x^8 y^5}$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $x^{-8} y^{-5}$.

Q: Can I use this problem in real-world applications?

A: Yes, you can use this problem in real-world applications such as simplifying expressions in algebra and calculus, solving equations and inequalities, and working with exponents in finance and economics.

Q: What are some common mistakes to avoid when multiplying expressions with exponents?

A: Some common mistakes to avoid when multiplying expressions with exponents include failing to follow the rules of exponent multiplication, not adding the exponents when the bases are the same, and not multiplying the expressions as they are when the bases are different.

Example Problems

• Find the product of $\left(x^4 y^3\right)\left(x^2 y^{-5}\right)$.
• Simplify the expression $x^{-6} y^4$.
• Find the product of $\left(x^9 y^{-15}\right)\left(x^{-12} y^{10}\right)$.

Solutions

• The product of $\left(x^4 y^3\right)\left(x^2 y^{-5}\right)$ is $x^6 y^{-2}$.
• The simplified expression $x^{-6} y^4$ is $\frac{1}{x^6 y^4}$.
• The product of $\left(x^9 y^{-15}\right)\left(x^{-12} y^{10}\right)$ is $x^{-8} y^{-5}$.

Conclusion

In conclusion, the product of $\left(x^9 y^{-15}\right)\left(x^{-12} y^{10}\right)$ is $x^{-8} y^{-5}$. We hope this Q&A article has helped you understand the rules of exponent multiplication and how to simplify expressions with exponents.