Simplify The Expression:$\left(x^4 Y^3\right)^3 \cdot (-2 X^2 Y^2$\]

Introduction

In this article, we will simplify the given expression $\left(x^4 y^3\right)^3 \cdot (-2 x^2 y^2)$. This involves applying the rules of exponents and simplifying the resulting expression. We will break down the process into manageable steps, making it easier to understand and follow along.

Understanding Exponents

Before we dive into simplifying the expression, let's review the rules of exponents. When we have a power raised to another power, we multiply the exponents. For example, $\left(x^2\right)^3 = x^{2 \cdot 3} = x^6$. This rule will be essential in simplifying the given expression.

Step 1: Apply the Power Rule

The first step is to apply the power rule to the expression $\left(x^4 y^3\right)^3$. Using the rule mentioned earlier, we multiply the exponents:

$$\left(x^4 y^3\right)^3 = x^{4 \cdot 3} y^{3 \cdot 3} = x^{12} y^9$$

Step 2: Multiply the Expression

Now that we have simplified the first part of the expression, we can multiply it by the second part, $(-2 x^2 y^2)$. When multiplying expressions with the same base, we add the exponents. However, in this case, we have a constant $-2$ that needs to be multiplied by the expression:

$$x^{12} y^9 \cdot (-2 x^2 y^2) = -2 x^{12 + 2} y^{9 + 2}$$

Step 3: Simplify the Exponents

Now we can simplify the exponents by adding them:

$$-2 x^{12 + 2} y^{9 + 2} = -2 x^{14} y^{11}$$

Step 4: Final Simplification

The expression is now simplified, but we can take it a step further by combining the constant $-2$ with the expression:

$$-2 x^{14} y^{11} = -2 x^{14} y^{11}$$

Conclusion

In this article, we simplified the expression $\left(x^4 y^3\right)^3 \cdot (-2 x^2 y^2)$ using the rules of exponents. We applied the power rule to simplify the first part of the expression, then multiplied it by the second part, and finally simplified the exponents. The resulting expression is $-2 x^{14} y^{11}$.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's essential to remember the following common mistakes:

• Not applying the power rule: Failing to apply the power rule can lead to incorrect simplifications.
• Not multiplying the exponents: When multiplying expressions with the same base, it's crucial to multiply the exponents.
• Not simplifying the exponents: Failing to simplify the exponents can result in an incorrect final expression.

Real-World Applications

Simplifying expressions with exponents has numerous real-world applications in various fields, including:

• Physics: Exponents are used to describe the behavior of physical systems, such as the motion of objects under the influence of gravity or friction.
• Engineering: Exponents are used to describe the behavior of electrical circuits, mechanical systems, and other complex systems.
• Computer Science: Exponents are used in algorithms and data structures to optimize performance and efficiency.

Final Thoughts

Introduction

In our previous article, we simplified the expression $\left(x^4 y^3\right)^3 \cdot (-2 x^2 y^2)$. In this article, we will answer some common questions related to simplifying expressions with exponents.

Q: What is the power rule in exponents?

A: The power rule in exponents states that when we have a power raised to another power, we multiply the exponents. For example, $\left(x^2\right)^3 = x^{2 \cdot 3} = x^6$.

Q: How do I apply the power rule to simplify an expression?

A: To apply the power rule, simply multiply the exponents. For example, if we have $\left(x^4 y^3\right)^3$, we would multiply the exponents as follows:

$$\left(x^4 y^3\right)^3 = x^{4 \cdot 3} y^{3 \cdot 3} = x^{12} y^9$$

Q: What happens when I multiply expressions with the same base?

A: When multiplying expressions with the same base, we add the exponents. For example, if we have $x^2 \cdot x^3$, we would add the exponents as follows:

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

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, we can rewrite the expression with a positive exponent by taking the reciprocal of the base. For example, if we have $x^{-2}$, we can rewrite it as follows:

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

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

A: Some common mistakes to avoid when simplifying expressions with exponents include:

• Not applying the power rule: Failing to apply the power rule can lead to incorrect simplifications.
• Not multiplying the exponents: When multiplying expressions with the same base, it's crucial to multiply the exponents.
• Not simplifying the exponents: Failing to simplify the exponents can result in an incorrect final expression.

Q: How do I apply the power rule to simplify expressions with multiple bases?

A: To apply the power rule to simplify expressions with multiple bases, we can use the following steps:

1. Simplify each base separately using the power rule.
2. Multiply the simplified bases together.

For example, if we have $\left(x^4 y^3\right)^3 \cdot (-2 x^2 y^2)$, we would simplify each base separately as follows:

$$\left(x^4 y^3\right)^3 = x^{4 \cdot 3} y^{3 \cdot 3} = x^{12} y^9$$

$$-2 x^2 y^2 = -2 x^2 y^2$$

Then, we would multiply the simplified bases together:

$$x^{12} y^9 \cdot (-2 x^2 y^2) = -2 x^{12 + 2} y^{9 + 2}$$

Q: What are some real-world applications of simplifying expressions with exponents?

A: Simplifying expressions with exponents has numerous real-world applications in various fields, including:

• Physics: Exponents are used to describe the behavior of physical systems, such as the motion of objects under the influence of gravity or friction.
• Engineering: Exponents are used to describe the behavior of electrical circuits, mechanical systems, and other complex systems.
• Computer Science: Exponents are used in algorithms and data structures to optimize performance and efficiency.

Conclusion

In this article, we answered some common questions related to simplifying expressions with exponents. We covered topics such as the power rule, multiplying expressions with the same base, and simplifying expressions with negative exponents. We also discussed common mistakes to avoid and real-world applications of simplifying expressions with exponents. By following the steps outlined in this article, you can simplify complex expressions and develop a deeper understanding of the underlying mathematics.