Simplify The Following Expression. Make Sure To Remove Any Negative Exponents.$\frac{\left(5 X^3 Y+6 X Y+5 X\right)^4\left(-3 X^3 Y-3 X+4\right)^2}{\left(5 X^3 Y+6 X Y+5 X\right)^6\left(4 X Y^2+2 Y^2+1\right)}$

Feb 27, 2025 by ADMIN 211 views

Simplify the Given Expression and Remove Negative Exponents

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Introduction

In this article, we will simplify the given expression and remove any negative exponents. The expression is a complex fraction involving multiple variables and exponents. We will break down the expression into smaller parts, simplify each part, and then combine them to obtain the final result.

Expression to Simplify

The given expression is:

$\frac{\left(5 x^3 y+6 x y+5 x\right)^4\left(-3 x^3 y-3 x+4\right)^2}{\left(5 x^3 y+6 x y+5 x\right)^6\left(4 x y^2+2 y^2+1\right)}$

Step 1: Factor Out Common Terms

To simplify the expression, we can start by factoring out common terms from the numerator and denominator.

Numerator

The numerator can be factored as:

$\left(5 x^3 y+6 x y+5 x\right)^4\left(-3 x^3 y-3 x+4\right)^2$

$= \left(5 x^3 y+6 x y+5 x\right)^4 \cdot \left(-3 x^3 y-3 x+4\right)^2$

Denominator

The denominator can be factored as:

$\left(5 x^3 y+6 x y+5 x\right)^6\left(4 x y^2+2 y^2+1\right)$

$= \left(5 x^3 y+6 x y+5 x\right)^6 \cdot \left(4 x y^2+2 y^2+1\right)$

Step 2: Simplify the Expression

Now that we have factored out common terms, we can simplify the expression by canceling out common factors.

Canceling Out Common Factors

We can cancel out the common factor $\left(5 x^3 y+6 x y+5 x\right)^4$ from the numerator and denominator.

$\frac{\left(5 x^3 y+6 x y+5 x\right)^4\left(-3 x^3 y-3 x+4\right)^2}{\left(5 x^3 y+6 x y+5 x\right)^6\left(4 x y^2+2 y^2+1\right)}$

$= \frac{\left(-3 x^3 y-3 x+4\right)^2}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

Step 3: Simplify the Expression Further

We can simplify the expression further by expanding the squared terms in the numerator.

Expanding Squared Terms

The squared term in the numerator can be expanded as:

$\left(-3 x^3 y-3 x+4\right)^2$

$= \left(-3 x^3 y\right)^2 + 2 \cdot \left(-3 x^3 y\right) \cdot \left(-3 x\right) + 2 \cdot \left(-3 x^3 y\right) \cdot 4 + \left(-3 x\right)^2 + 2 \cdot \left(-3 x\right) \cdot 4 + 4^2$

$= 9 x^6 y^2 + 18 x^4 y + 24 x^3 y - 9 x^2 - 24 x + 16$

Step 4: Simplify the Expression Further

We can simplify the expression further by canceling out common factors.

Canceling Out Common Factors

We can cancel out the common factor $9 x^2$ from the numerator and denominator.

$\frac{9 x^6 y^2 + 18 x^4 y + 24 x^3 y - 9 x^2 - 24 x + 16}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

$= \frac{9 x^4 y^2 + 18 x^2 y + 24 x y - 1 - \frac{24}{9 x^2} + \frac{16}{9 x^2}}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

Step 5: Remove Negative Exponents

We can remove the negative exponents by multiplying the numerator and denominator by the reciprocal of the negative exponent.

Removing Negative Exponents

We can remove the negative exponent $\frac{24}{9 x^2}$ by multiplying the numerator and denominator by $x^2$ .

$\frac{9 x^4 y^2 + 18 x^2 y + 24 x y - 1 - \frac{24}{9 x^2} + \frac{16}{9 x^2}}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

$= \frac{9 x^4 y^2 + 18 x^2 y + 24 x y - 1 - \frac{8}{3 x^2} + \frac{16}{9 x^2}}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

Step 6: Simplify the Expression Further

We can simplify the expression further by combining like terms.

Combining Like Terms

We can combine the like terms in the numerator.

$\frac{9 x^4 y^2 + 18 x^2 y + 24 x y - 1 - \frac{8}{3 x^2} + \frac{16}{9 x^2}}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

$= \frac{9 x^4 y^2 + 18 x^2 y + 24 x y - 1 - \frac{8}{3 x^2} + \frac{16}{9 x^2}}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

Step 7: Final Simplification

We can simplify the expression further by canceling out common factors.

Canceling Out Common Factors

We can cancel out the common factor $9 x^2$ from the numerator and denominator.

$\frac{9 x^4 y^2 + 18 x^2 y + 24 x y - 1 - \frac{8}{3 x^2} + \frac{16}{9 x^2}}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

$= \frac{9 x^2 y^2 + 18 y + 24 y - \frac{8}{3} + \frac{16}{9 x^2}}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

Conclusion

In this article, we simplified the given expression and removed any negative exponents. We broke down the expression into smaller parts, simplified each part, and then combined them to obtain the final result. The final simplified expression is:

$\frac{9 x^2 y^2 + 18 y + 24 y - \frac{8}{3} + \frac{16}{9 x^2}}{\left(5 x^3 y+6 x y+5 x\right)^2\left(4 x y^2+2 y^2+1\right)}$

This expression is the final simplified result of the given expression.

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Introduction

In our previous article, we simplified the given expression and removed any negative exponents. In this article, we will answer some common questions related to the simplification process.

Q: What is the purpose of simplifying an expression?

A: The purpose of simplifying an expression is to make it easier to understand and work with. Simplifying an expression can help to:

Reduce the complexity of the expression
Make it easier to identify patterns and relationships
Improve the accuracy of calculations
Make it easier to solve problems and equations

Q: What are some common techniques used to simplify expressions?

A: Some common techniques used to simplify expressions include:

Factoring: breaking down an expression into simpler components
Canceling: canceling out common factors between the numerator and denominator
Combining like terms: combining terms with the same variable and exponent
Simplifying fractions: simplifying fractions by canceling out common factors

Q: How do I know when to use each technique?

A: The choice of technique depends on the specific expression and the goal of the simplification process. Here are some general guidelines:

Use factoring when the expression can be broken down into simpler components
Use canceling when there are common factors between the numerator and denominator
Use combining like terms when there are terms with the same variable and exponent
Use simplifying fractions when the fraction can be simplified by canceling out common factors

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

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

Not canceling out common factors
Not combining like terms
Not simplifying fractions
Not checking for errors in the simplification process

Q: How do I check for errors in the simplification process?

A: To check for errors in the simplification process, follow these steps:

Review the original expression and the simplified expression
Check for any errors in the simplification process
Verify that the simplified expression is equivalent to the original expression
Check for any errors in the cancellation process

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

A: Simplifying expressions has many real-world applications, including:

Calculating mathematical functions and formulas
Solving problems and equations in physics, engineering, and other fields
Analyzing data and making predictions
Developing algorithms and models for complex systems

Q: How can I practice simplifying expressions?

A: To practice simplifying expressions, try the following:

Start with simple expressions and gradually move to more complex ones
Practice simplifying expressions with different variables and exponents
Use online resources and tools to help with the simplification process
Work with a partner or tutor to get feedback and guidance

Conclusion

In this article, we answered some common questions related to simplifying expressions and removing negative exponents. We covered topics such as the purpose of simplifying expressions, common techniques used to simplify expressions, and common mistakes to avoid. We also discussed how to check for errors in the simplification process and provided some real-world applications of simplifying expressions. Finally, we provided some tips for practicing simplifying expressions.

Additional Resources

For more information on simplifying expressions and removing negative exponents, check out the following resources:

Online tutorials and videos
Math textbooks and workbooks
Online communities and forums
Math software and calculators

By following these resources and practicing simplifying expressions, you can become more confident and proficient in simplifying expressions and removing negative exponents.