Use The Product Of Powers Property To Simplify The Expression \[$-2 X^3 Y^4 X^2 Y^3\$\].A. \[$-2(x Y)^{12}\$\]B. \[$-2(x Y)^{72}\$\]C. \[$-2 X^6 Y^{12}\$\]D. \[$-2 X^5 Y^7\$\]Note: The Product Of Powers Property

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. One of the key properties used to simplify expressions is the product of powers property. This property states that when we multiply two powers with the same base, we can add their exponents. In this article, we will explore how to use the product of powers property to simplify the expression ${-2 x^3 y^4 x^2 y^3}$.

Understanding the Product of Powers Property

The product of powers property is a fundamental concept in algebra that helps us simplify expressions by combining powers with the same base. The property states that when we multiply two powers with the same base, we can add their exponents. Mathematically, this can be represented as:

${a^m \cdot a^n = a^{m+n}}$

where ${a}$ is the base and ${m}$ and ${n}$ are the exponents.

Applying the Product of Powers Property to the Given Expression

Now that we have a good understanding of the product of powers property, let's apply it to the given expression ${-2 x^3 y^4 x^2 y^3}$. To simplify this expression, we need to combine the powers of ${x}$ and ${y}$ using the product of powers property.

Combining Powers of x

We can start by combining the powers of ${x}$ in the expression. We have two powers of ${x}$, ${x^3}$ and ${x^2}$, which we can combine using the product of powers property:

${x^3 \cdot x^2 = x^{3+2} = x^5}$

So, the expression ${-2 x^3 y^4 x^2 y^3}$ can be simplified to ${-2 x^5 y^4 y^3}$.

Combining Powers of y

Next, we can combine the powers of ${y}$ in the expression. We have two powers of ${y}$, ${y^4}$ and ${y^3}$, which we can combine using the product of powers property:

${y^4 \cdot y^3 = y^{4+3} = y^7}$

So, the expression ${-2 x^5 y^4 y^3}$ can be simplified to ${-2 x^5 y^7}$.

Final Simplified Expression

Therefore, the final simplified expression is ${-2 x^5 y^7}$.

Conclusion

In this article, we used the product of powers property to simplify the expression ${-2 x^3 y^4 x^2 y^3}$. We combined the powers of ${x}$ and ${y}$ using the product of powers property and arrived at the final simplified expression ${-2 x^5 y^7}$. This example demonstrates the importance of the product of powers property in simplifying algebraic expressions.

Answer

The correct answer is:

{-2 x^5 y^7$}$

Note

Introduction

In our previous article, we explored the product of powers property and how it can be used to simplify algebraic expressions. In this article, we will answer some frequently asked questions (FAQs) about the product of powers property. Whether you're a math student or a professional, these FAQs will help you understand the product of powers property and how to apply it in different situations.

Q: What is the product of powers property?

A: The product of powers property is a fundamental concept in algebra that states that when we multiply two powers with the same base, we can add their exponents. Mathematically, this can be represented as:

${a^m \cdot a^n = a^{m+n}}$

where ${a}$ is the base and ${m}$ and ${n}$ are the exponents.

Q: How do I apply the product of powers property?

A: To apply the product of powers property, you need to identify the powers with the same base in the expression. Then, you can add their exponents and simplify the expression.

For example, consider the expression ${x^3 \cdot x^2}$. To simplify this expression, you can add the exponents of ${x}$:

${x^3 \cdot x^2 = x^{3+2} = x^5}$

Q: What if I have multiple powers with the same base?

A: If you have multiple powers with the same base, you can add their exponents and simplify the expression. For example, consider the expression ${x^3 \cdot x^2 \cdot x^4}$. To simplify this expression, you can add the exponents of ${x}$:

${x^3 \cdot x^2 \cdot x^4 = x^{3+2+4} = x^9}$

Q: Can I apply the product of powers property to expressions with different bases?

A: No, the product of powers property only applies to expressions with the same base. If you have expressions with different bases, you cannot apply the product of powers property.

For example, consider the expression ${x^3 \cdot y^2}$. In this expression, ${x}$ and ${y}$ are different bases, so you cannot apply the product of powers property.

Q: What if I have a negative exponent?

A: If you have a negative exponent, you can still apply the product of powers property. For example, consider the expression ${x^{-3} \cdot x^2}$. To simplify this expression, you can add the exponents of ${x}$:

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

Q: Can I apply the product of powers property to expressions with fractions?

A: Yes, you can apply the product of powers property to expressions with fractions. For example, consider the expression ${\frac{x^3}{x^2}}$. To simplify this expression, you can subtract the exponents of ${x}$:

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

Conclusion

In this article, we answered some frequently asked questions (FAQs) about the product of powers property. Whether you're a math student or a professional, these FAQs will help you understand the product of powers property and how to apply it in different situations.

Remember

The product of powers property is a fundamental concept in algebra that helps you simplify expressions by combining powers with the same base. With practice and patience, you can master the product of powers property and become proficient in simplifying algebraic expressions.

Practice Problems

To practice the product of powers property, try the following problems:

1. Simplify the expression ${x^3 \cdot x^2}$.
2. Simplify the expression ${x^3 \cdot x^2 \cdot x^4}$.
3. Simplify the expression ${x^{-3} \cdot x^2}$.
4. Simplify the expression ${\frac{x^3}{x^2}}$.

Answer Key

1. ${x^5}$
2. ${x^9}$
3. ${x^{-1}}$
4. ${x^1}$