Simplify The Expression:${ \frac{10 X^3 Y^2}{35 X^2 Y^4} \div \frac{2 X^2 Y^2}{7 X^4 Y^2} }$

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements with ease. In this article, we will focus on simplifying a given expression involving variables and fractions. We will break down the problem into manageable steps, using algebraic properties and techniques to arrive at the final simplified expression.

The Given Expression

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The given expression is:

$$\frac{10 x^3 y^2}{35 x^2 y^4} \div \frac{2 x^2 y^2}{7 x^4 y^2}$$

This expression involves fractions, variables, and exponents. Our goal is to simplify it by applying algebraic properties and techniques.

Step 1: Simplify the First Fraction

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To simplify the first fraction, we can start by factoring out common factors from the numerator and denominator.

$$\frac{10 x^3 y^2}{35 x^2 y^4} = \frac{2 \cdot 5 \cdot x^3 \cdot y^2}{7 \cdot 5 \cdot x^2 \cdot y^4}$$

We can cancel out the common factor of 5 from the numerator and denominator, as well as the common factor of $x^2$.

$$\frac{2 \cdot x \cdot y^2}{7 \cdot y^4}$$

Step 2: Simplify the Second Fraction

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Next, we can simplify the second fraction by factoring out common factors from the numerator and denominator.

$$\frac{2 x^2 y^2}{7 x^4 y^2} = \frac{2 \cdot x^2 \cdot y^2}{7 \cdot x^4 \cdot y^2}$$

We can cancel out the common factor of $x^2$ from the numerator and denominator, as well as the common factor of $y^2$.

$$\frac{2}{7 \cdot x^2}$$

Step 3: Divide the Fractions

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Now that we have simplified both fractions, we can divide them by multiplying the first fraction by the reciprocal of the second fraction.

$$\frac{2 \cdot x \cdot y^2}{7 \cdot y^4} \div \frac{2}{7 \cdot x^2} = \frac{2 \cdot x \cdot y^2}{7 \cdot y^4} \cdot \frac{7 \cdot x^2}{2}$$

We can cancel out the common factor of 2 from the numerator and denominator, as well as the common factor of 7.

$$\frac{x \cdot y^2}{y^4} \cdot x^2$$

Step 4: Simplify the Expression

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Finally, we can simplify the expression by combining like terms and applying exponent rules.

$$\frac{x \cdot y^2}{y^4} \cdot x^2 = \frac{1}{y^2} \cdot x^3$$

We can rewrite the expression in a more simplified form by combining the variables.

$$\frac{x^3}{y^2}$$

Conclusion

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In this article, we simplified a given expression involving variables and fractions by applying algebraic properties and techniques. We broke down the problem into manageable steps, factoring out common factors, canceling out common terms, and multiplying fractions. By following these steps, we arrived at the final simplified expression.

Tips and Tricks

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• When simplifying expressions, it's essential to identify common factors and cancel them out.
• When dividing fractions, multiply the first fraction by the reciprocal of the second fraction.
• When combining like terms, apply exponent rules and simplify the expression.

Practice Problems

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1. Simplify the expression: $\frac{3 x^2 y^3}{6 x^3 y^2} \div \frac{x y^2}{3 x^2 y}$
2. Simplify the expression: $\frac{2 x^3 y^4}{4 x^2 y^3} \div \frac{x^2 y^2}{2 x y}$
3. Simplify the expression: $\frac{5 x^2 y^3}{10 x^3 y^2} \div \frac{x y^2}{5 x^2 y}$

Final Thoughts

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Simplifying expressions is a crucial skill in algebra that helps us solve equations and manipulate mathematical statements with ease. By applying algebraic properties and techniques, we can simplify complex expressions and arrive at the final answer. Remember to identify common factors, cancel them out, and multiply fractions when dividing. With practice and patience, you'll become proficient in simplifying expressions and tackling complex algebraic problems.

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Introduction

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In our previous article, we simplified a given expression involving variables and fractions by applying algebraic properties and techniques. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used in simplifying expressions.

Q&A

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Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to identify common factors in the numerator and denominator. We can then cancel out these common factors to simplify the expression.

Q: How do I simplify a fraction?

A: To simplify a fraction, we can factor out common factors from the numerator and denominator. We can then cancel out these common factors to simplify the fraction.

Q: What is the difference between simplifying a fraction and dividing fractions?

A: Simplifying a fraction involves factoring out common factors and canceling them out, whereas dividing fractions involves multiplying the first fraction by the reciprocal of the second fraction.

Q: How do I divide fractions?

A: To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. This means we flip the second fraction and multiply the numerators and denominators.

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

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

• Not identifying common factors
• Not canceling out common factors
• Not multiplying fractions correctly
• Not simplifying the expression fully

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through practice problems and exercises. You can also try simplifying expressions on your own and then checking your work with a calculator or a friend.

Tips and Tricks

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• When simplifying expressions, it's essential to identify common factors and cancel them out.
• When dividing fractions, multiply the first fraction by the reciprocal of the second fraction.
• When combining like terms, apply exponent rules and simplify the expression.
• Always check your work to ensure that the expression is fully simplified.

Common Mistakes

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• Not identifying common factors
• Not canceling out common factors
• Not multiplying fractions correctly
• Not simplifying the expression fully

Practice Problems

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1. Simplify the expression: $\frac{3 x^2 y^3}{6 x^3 y^2} \div \frac{x y^2}{3 x^2 y}$
2. Simplify the expression: $\frac{2 x^3 y^4}{4 x^2 y^3} \div \frac{x^2 y^2}{2 x y}$
3. Simplify the expression: $\frac{5 x^2 y^3}{10 x^3 y^2} \div \frac{x y^2}{5 x^2 y}$

Final Thoughts

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Simplifying expressions is a crucial skill in algebra that helps us solve equations and manipulate mathematical statements with ease. By applying algebraic properties and techniques, we can simplify complex expressions and arrive at the final answer. Remember to identify common factors, cancel them out, and multiply fractions when dividing. With practice and patience, you'll become proficient in simplifying expressions and tackling complex algebraic problems.

Additional Resources

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• Algebra textbooks and online resources
• Practice problems and exercises
• Online calculators and math software
• Math tutors and online communities

Conclusion

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In this article, we provided a Q&A guide to help you understand the concepts and techniques used in simplifying expressions. We covered common mistakes to avoid, tips and tricks for simplifying expressions, and practice problems to help you practice your skills. Remember to always check your work and simplify expressions fully to ensure that you arrive at the correct answer. With practice and patience, you'll become proficient in simplifying expressions and tackling complex algebraic problems.