Simplify The Expression:$ \frac{\frac{(x+y) {-2}}{y {-3}}}{\frac{x^3 Y {-2}(x+y) {-1}}{x^{-2}}} }$Choose The Correct Simplification A. { \frac{y^5 {x^6 + X^5 Y}$}$B. { \frac{x+y}{x Y}$} C . \[ C. \[ C . \[ \frac{y 5}{x 2 + X

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Introduction

Algebraic expressions can be complex and daunting, especially when they involve multiple variables and operations. Simplifying these expressions is a crucial skill in mathematics, as it allows us to better understand and work with them. In this article, we will focus on simplifying a specific complex algebraic expression, step by step.

The Given Expression

The expression we will be simplifying is:

(x+y)−2y−3x3y−2(x+y)−1x−2\frac{\frac{(x+y)^{-2}}{y^{-3}}}{\frac{x^3 y^{-2}(x+y)^{-1}}{x^{-2}}}

This expression involves multiple variables, exponents, and fractions. Our goal is to simplify it to its most basic form.

Step 1: Simplify the Numerator

Let's start by simplifying the numerator of the expression. We have:

(x+y)−2y−3\frac{(x+y)^{-2}}{y^{-3}}

Using the rule for dividing exponents with the same base, we can rewrite this as:

(x+y)−2y−3=(x+y)−2y3\frac{(x+y)^{-2}}{y^{-3}} = (x+y)^{-2}y^3

Step 2: Simplify the Denominator

Next, let's simplify the denominator of the expression. We have:

x3y−2(x+y)−1x−2\frac{x^3 y^{-2}(x+y)^{-1}}{x^{-2}}

Using the rule for dividing exponents with the same base, we can rewrite this as:

x3y−2(x+y)−1x−2=x5y−2(x+y)−1\frac{x^3 y^{-2}(x+y)^{-1}}{x^{-2}} = x^5 y^{-2}(x+y)^{-1}

Step 3: Combine the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the simplified expression:

(x+y)−2y3x5y−2(x+y)−1\frac{(x+y)^{-2}y^3}{x^5 y^{-2}(x+y)^{-1}}

Step 4: Simplify the Expression

To simplify the expression further, we can cancel out common factors in the numerator and denominator. We have:

(x+y)−2y3x5y−2(x+y)−1=y5x5(x+y)\frac{(x+y)^{-2}y^3}{x^5 y^{-2}(x+y)^{-1}} = \frac{y^5}{x^5(x+y)}

Step 5: Final Simplification

Finally, we can simplify the expression by combining the terms in the denominator:

y5x5(x+y)=y5x5y+x5\frac{y^5}{x^5(x+y)} = \frac{y^5}{x^5y + x^5}

Conclusion

In conclusion, the simplified expression is:

y5x5y+x5\frac{y^5}{x^5y + x^5}

This is the final answer to the given problem.

Discussion

The given expression can be simplified in different ways, depending on the approach taken. However, the correct simplification is the one that results in the expression:

y5x5y+x5\frac{y^5}{x^5y + x^5}

This expression can be further simplified by factoring out common terms:

y5x5y+x5=y5x5(y+1)\frac{y^5}{x^5y + x^5} = \frac{y^5}{x^5(y + 1)}

This is the final simplified form of the expression.

Comparison with Options

Let's compare the simplified expression with the given options:

A. y5x6+x5y\frac{y^5}{x^6 + x^5 y}

B. x+yxy\frac{x+y}{x y}

C. y5x2+x\frac{y^5}{x^2 + x}

The correct simplification is option A, which is:

y5x6+x5y\frac{y^5}{x^6 + x^5 y}

This is the same as the simplified expression we obtained earlier:

y5x5y+x5\frac{y^5}{x^5y + x^5}

Conclusion

In conclusion, the correct simplification of the given expression is:

y5x6+x5y\frac{y^5}{x^6 + x^5 y}

This is the final answer to the problem.

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Introduction

In our previous article, we simplified a complex algebraic expression step by step. In this article, we will answer some frequently asked questions related to simplifying complex algebraic expressions.

Q&A

Q: What are some common mistakes to avoid when simplifying complex algebraic expressions?

A: Some common mistakes to avoid when simplifying complex algebraic expressions include:

  • Not following the order of operations (PEMDAS)
  • Not simplifying fractions correctly
  • Not canceling out common factors
  • Not combining like terms

Q: How do I simplify a complex algebraic expression with multiple variables?

A: To simplify a complex algebraic expression with multiple variables, follow these steps:

  1. Identify the variables and their exponents
  2. Simplify the expression using the rules of exponents
  3. Combine like terms
  4. Cancel out common factors

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves reducing it to its most basic form, while solving an equation involves finding the value of the variable that makes the equation true.

Q: How do I know when to simplify an expression?

A: You should simplify an expression when:

  • It is necessary to solve an equation or inequality
  • It is necessary to compare two expressions
  • It is necessary to evaluate an expression

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by rewriting it as a positive exponent. For example:

1x−2=x2\frac{1}{x^{-2}} = x^2

Q: How do I simplify an expression with a fraction in the denominator?

A: To simplify an expression with a fraction in the denominator, follow these steps:

  1. Simplify the fraction in the denominator
  2. Cancel out common factors
  3. Combine like terms

Q: Can I simplify an expression with a variable in the denominator?

A: Yes, you can simplify an expression with a variable in the denominator by rewriting it as a fraction with a variable in the numerator. For example:

1x=x0x=1x\frac{1}{x} = \frac{x^0}{x} = \frac{1}{x}

Q: How do I simplify an expression with multiple fractions?

A: To simplify an expression with multiple fractions, follow these steps:

  1. Simplify each fraction separately
  2. Combine like terms
  3. Cancel out common factors

Conclusion

In conclusion, simplifying complex algebraic expressions requires attention to detail and a clear understanding of the rules of exponents and fractions. By following the steps outlined in this article, you can simplify complex algebraic expressions with ease.

Additional Resources

For more information on simplifying complex algebraic expressions, check out the following resources:

  • Khan Academy: Simplifying Algebraic Expressions
  • Mathway: Simplifying Algebraic Expressions
  • Wolfram Alpha: Simplifying Algebraic Expressions

Discussion

Simplifying complex algebraic expressions is an essential skill in mathematics. By practicing and mastering this skill, you can solve equations and inequalities with ease. Do you have any questions or topics you would like to discuss?