Simplify The Expression:${ \left(\frac{-4 X^4 Y^{-2}}{2 X^5 Y {-4}}\right) 2 \cdot \left(5 X Y 2\right) 2 }$

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying a given expression, which involves exponentiation, multiplication, and division of variables with exponents. Our goal is to break down the expression into manageable parts, apply the rules of exponents, and finally simplify the resulting expression.

Understanding the Expression

The given expression is:

$$\left(\frac{-4 x^4 y^{-2}}{2 x^5 y^{-4}}\right)^2 \cdot \left(5 x y^2\right)^2$$

This expression involves two main components:

1. The fraction $\frac{-4 x^4 y^{-2}}{2 x^5 y^{-4}}$ raised to the power of 2.
2. The term $5 x y^2$ raised to the power of 2.

Breaking Down the Expression

To simplify the expression, we need to break it down into smaller parts and apply the rules of exponents. Let's start by simplifying the fraction:

$$\frac{-4 x^4 y^{-2}}{2 x^5 y^{-4}}$$

We can simplify this fraction by applying the rule of dividing like bases with exponents:

$$\frac{-4 x^4 y^{-2}}{2 x^5 y^{-4}} = \frac{-4}{2} \cdot x^{4-5} \cdot y^{-2-(-4)}$$

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

Now, let's simplify the term $5 x y^2$ raised to the power of 2:

$$(5 x y^2)^2 = 25 x^2 y^4$$

Applying the Rules of Exponents

Now that we have simplified the fraction and the term, we can apply the rules of exponents to simplify the expression. When multiplying variables with exponents, we add the exponents:

$$(-2 \cdot x^{-1} \cdot y^2)^2 \cdot 25 x^2 y^4$$

$$= 4 x^{-2} y^4 \cdot 25 x^2 y^4$$

$$= 100 x^{-2+2} y^{4+4}$$

$$= 100 x^0 y^8$$

Simplifying the Expression

Now that we have applied the rules of exponents, we can simplify the expression further. When the exponent of a variable is 0, the variable is equal to 1:

$$100 x^0 y^8 = 100 \cdot 1 \cdot y^8$$

$$= 100 y^8$$

Conclusion

In this article, we simplified the given expression by breaking it down into manageable parts, applying the rules of exponents, and finally simplifying the resulting expression. We started by simplifying the fraction and the term, then applied the rules of exponents to simplify the expression. The final simplified expression is $100 y^8$. This example demonstrates the importance of algebraic manipulation in mathematics and highlights the need for a clear understanding of the rules of exponents.

Frequently Asked Questions

• What is the rule for dividing like bases with exponents?
  • When dividing like bases with exponents, we subtract the exponents.
• What is the rule for multiplying variables with exponents?
  • When multiplying variables with exponents, we add the exponents.
• What is the rule for simplifying an expression with a variable raised to the power of 0?
  • When the exponent of a variable is 0, the variable is equal to 1.

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires a clear understanding of the rules of exponents. By breaking down the expression into manageable parts, applying the rules of exponents, and finally simplifying the resulting expression, we can simplify even the most complex expressions. This article provides a step-by-step guide to simplifying the given expression, and it highlights the importance of algebraic manipulation in mathematics.

Introduction

In our previous article, we simplified the given expression by breaking it down into manageable parts, applying the rules of exponents, and finally simplifying the resulting expression. In this article, we will provide a Q&A section to address some of the most frequently asked questions related to simplifying expressions.

Q&A

Q: What is the rule for simplifying an expression with a variable raised to the power of 0?

A: When the exponent of a variable is 0, the variable is equal to 1. For example, $x^0 = 1$.

Q: What is the rule for dividing like bases with exponents?

A: When dividing like bases with exponents, we subtract the exponents. For example, $\frac{x^4}{x^2} = x^{4-2} = x^2$.

Q: What is the rule for multiplying variables with exponents?

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

Q: How do I simplify an expression with multiple variables and exponents?

A: To simplify an expression with multiple variables and exponents, you need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, while a constant is a value that remains the same.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you need to follow the rule:

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

For example, $x^{-2} = \frac{1}{x^2}$.

Q: What is the rule for simplifying an expression with a fraction raised to a power?

A: To simplify an expression with a fraction raised to a power, you need to follow the rule:

$$(\frac{a}{b})^n = \frac{a^n}{b^n}$$

For example, $(\frac{2}{3})^2 = \frac{2^2}{3^2} = \frac{4}{9}$.

Conclusion

In this article, we provided a Q&A section to address some of the most frequently asked questions related to simplifying expressions. We covered topics such as simplifying expressions with variables raised to the power of 0, dividing like bases with exponents, multiplying variables with exponents, and simplifying expressions with multiple variables and exponents. By following the rules of exponents and the order of operations, you can simplify even the most complex expressions.

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires a clear understanding of the rules of exponents. By breaking down the expression into manageable parts, applying the rules of exponents, and finally simplifying the resulting expression, we can simplify even the most complex expressions. This article provides a step-by-step guide to simplifying the given expression, and it highlights the importance of algebraic manipulation in mathematics.

Additional Resources

• Khan Academy: Algebraic Manipulation
• Mathway: Algebraic Manipulation
• Wolfram Alpha: Algebraic Manipulation

Frequently Asked Questions

• What is the rule for simplifying an expression with a variable raised to the power of 0?
• What is the rule for dividing like bases with exponents?
• What is the rule for multiplying variables with exponents?
• How do I simplify an expression with multiple variables and exponents?
• What is the difference between a variable and a constant?
• How do I simplify an expression with a negative exponent?
• What is the rule for simplifying an expression with a fraction raised to a power?