Simplify The Expression:${ \frac{3 X^2 Y^4}{4 X^{-5}} \times \frac{8 X^2 Y^{-3}}{9 Y^{-1}} }$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill that every student and mathematician should possess. In this article, we will delve into the world of algebraic manipulation and provide a step-by-step guide on how to simplify the given expression: 3x2y44x−5×8x2y−39y−1\frac{3 x^2 y^4}{4 x^{-5}} \times \frac{8 x^2 y^{-3}}{9 y^{-1}}. We will explore the rules of exponents, the concept of like terms, and the importance of simplifying expressions in mathematics.

Understanding the Rules of Exponents

Before we dive into the simplification process, it's essential to understand the rules of exponents. Exponents are a shorthand way of representing repeated multiplication of a number. For example, x2x^2 means xx multiplied by itself twice, or x×xx \times x. The rules of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, x2×x3=x2+3=x5x^2 \times x^3 = x^{2+3} = x^5.

Simplifying the Expression

Now that we have a solid understanding of the rules of exponents, let's simplify the given expression. The expression is a product of two fractions, and we can simplify it by multiplying the numerators and denominators separately.

3x2y44x−5×8x2y−39y−1\frac{3 x^2 y^4}{4 x^{-5}} \times \frac{8 x^2 y^{-3}}{9 y^{-1}}

To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Multiply the numerators: 3×8=243 \times 8 = 24
  2. Multiply the denominators: 4×9=364 \times 9 = 36
  3. Simplify the exponents: x−5×x2=x−5+2=x−3x^{-5} \times x^2 = x^{-5+2} = x^{-3}
  4. Simplify the exponents: y4×y−3=y4−3=y1y^4 \times y^{-3} = y^{4-3} = y^1
  5. Simplify the exponents: y−1×y−3=y−1−3=y−4y^{-1} \times y^{-3} = y^{-1-3} = y^{-4}

Now that we have simplified the exponents, we can rewrite the expression as:

24x−3y136y−4\frac{24 x^{-3} y^1}{36 y^{-4}}

Canceling Out Like Terms

The next step in simplifying the expression is to cancel out like terms. Like terms are terms that have the same variable and exponent. In this case, we can cancel out the y1y^1 term in the numerator with the y−4y^{-4} term in the denominator.

24x−3y136y−4\frac{24 x^{-3} y^1}{36 y^{-4}}

To cancel out the like terms, we need to multiply the numerator and denominator by the reciprocal of the denominator's exponent. In this case, we need to multiply by y4y^4.

24x−3y1×y436y−4×y4\frac{24 x^{-3} y^1 \times y^4}{36 y^{-4} \times y^4}

Now that we have canceled out the like terms, we can simplify the expression further.

24x−3y536\frac{24 x^{-3} y^5}{36}

Final Simplification

The final step in simplifying the expression is to simplify the fraction. We can do this by dividing the numerator by the denominator.

24x−3y536=2x−3y53\frac{24 x^{-3} y^5}{36} = \frac{2 x^{-3} y^5}{3}

And that's it! We have successfully simplified the given expression.

Conclusion

Simplifying algebraic expressions is a crucial skill that every student and mathematician should possess. In this article, we have provided a step-by-step guide on how to simplify the given expression: 3x2y44x−5×8x2y−39y−1\frac{3 x^2 y^4}{4 x^{-5}} \times \frac{8 x^2 y^{-3}}{9 y^{-1}}. We have explored the rules of exponents, the concept of like terms, and the importance of simplifying expressions in mathematics. By following the order of operations and canceling out like terms, we have successfully simplified the expression.

Frequently Asked Questions

  • What are the rules of exponents? The rules of exponents state that when we multiply two numbers with the same base, we add their exponents.
  • How do we simplify expressions with like terms? We can simplify expressions with like terms by canceling them out. To do this, we need to multiply the numerator and denominator by the reciprocal of the denominator's exponent.
  • Why is it important to simplify expressions in mathematics? Simplifying expressions in mathematics is important because it helps us to understand the underlying structure of the expression and to make it easier to work with.

Final Thoughts

Simplifying algebraic expressions is a crucial skill that every student and mathematician should possess. By following the rules of exponents and canceling out like terms, we can simplify even the most complex expressions. In this article, we have provided a step-by-step guide on how to simplify the given expression: 3x2y44x−5×8x2y−39y−1\frac{3 x^2 y^4}{4 x^{-5}} \times \frac{8 x^2 y^{-3}}{9 y^{-1}}. We hope that this article has been helpful in providing a comprehensive guide to algebraic manipulation.

Introduction

Algebraic manipulation is a crucial skill that every student and mathematician should possess. In our previous article, we provided a comprehensive guide on how to simplify the given expression: 3x2y44x−5×8x2y−39y−1\frac{3 x^2 y^4}{4 x^{-5}} \times \frac{8 x^2 y^{-3}}{9 y^{-1}}. In this article, we will provide a Q&A guide to help you better understand the concepts of algebraic manipulation.

Q&A Guide

Q: What are the rules of exponents?

A: The rules of exponents state that when we multiply two numbers with the same base, we add their exponents. For example, x2×x3=x2+3=x5x^2 \times x^3 = x^{2+3} = x^5.

Q: How do we simplify expressions with like terms?

A: We can simplify expressions with like terms by canceling them out. To do this, we need to multiply the numerator and denominator by the reciprocal of the denominator's exponent.

Q: Why is it important to simplify expressions in mathematics?

A: Simplifying expressions in mathematics is important because it helps us to understand the underlying structure of the expression and to make it easier to work with.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

  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: How do we handle negative exponents?

A: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, x−3=1x3x^{-3} = \frac{1}{x^3}.

Q: Can we simplify expressions with variables in the denominator?

A: Yes, we can simplify expressions with variables in the denominator by multiplying the numerator and denominator by the reciprocal of the denominator's exponent.

Q: How do we handle fractions with variables in the numerator and denominator?

A: We can handle fractions with variables in the numerator and denominator by simplifying the expression by canceling out like terms.

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 does not change.

Q: How do we simplify expressions with multiple variables?

A: We can simplify expressions with multiple variables by using the rules of exponents and canceling out like terms.

Conclusion

Algebraic manipulation is a crucial skill that every student and mathematician should possess. In this article, we have provided a Q&A guide to help you better understand the concepts of algebraic manipulation. We hope that this guide has been helpful in providing a comprehensive overview of algebraic manipulation.

Frequently Asked Questions

  • What are the rules of exponents?
  • How do we simplify expressions with like terms?
  • Why is it important to simplify expressions in mathematics?
  • What is the order of operations?
  • How do we handle negative exponents?
  • Can we simplify expressions with variables in the denominator?
  • How do we handle fractions with variables in the numerator and denominator?
  • What is the difference between a variable and a constant?
  • How do we simplify expressions with multiple variables?

Final Thoughts

Algebraic manipulation is a crucial skill that every student and mathematician should possess. By understanding the rules of exponents, simplifying expressions with like terms, and following the order of operations, we can simplify even the most complex expressions. We hope that this Q&A guide has been helpful in providing a comprehensive overview of algebraic manipulation.