Simplify The Expression. Rewrite With No Negative Exponents.${ \frac{10 X^2 Y^{-4}}{5 X^{-3} Y^{-1}} }$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One common challenge when simplifying expressions is dealing with negative exponents. In this article, we will explore how to simplify the given expression by rewriting it with no negative exponents.

The Given Expression

The given expression is:

10x2yβˆ’45xβˆ’3yβˆ’1{ \frac{10 x^2 y^{-4}}{5 x^{-3} y^{-1}} }

This expression involves both positive and negative exponents, which can make it difficult to simplify. Our goal is to rewrite this expression with no negative exponents.

Understanding Negative Exponents

Before we dive into simplifying the expression, let's take a moment to understand what negative exponents represent. A negative exponent indicates that the base is being raised to a power that is the reciprocal of the given exponent. In other words, a negative exponent is equivalent to taking the reciprocal of the base and raising it to the positive exponent.

For example, the expression xβˆ’2x^{-2} is equivalent to 1x2\frac{1}{x^2}.

Simplifying the Expression

To simplify the given expression, we will use the following steps:

  1. Invert and Multiply: We will invert the second fraction and multiply it by the first fraction.
  2. Combine Like Terms: We will combine the like terms in the numerator and denominator.
  3. Simplify the Expression: We will simplify the resulting expression by canceling out any common factors.

Step 1: Invert and Multiply

To invert the second fraction, we will multiply it by the reciprocal of the denominator. This will give us:

10x2yβˆ’45xβˆ’3yβˆ’1Γ—115xβˆ’3yβˆ’1{ \frac{10 x^2 y^{-4}}{5 x^{-3} y^{-1}} \times \frac{1}{\frac{1}{5 x^{-3} y^{-1}}} }

Simplifying this expression, we get:

10x2yβˆ’45xβˆ’3yβˆ’1Γ—5xβˆ’3yβˆ’11{ \frac{10 x^2 y^{-4}}{5 x^{-3} y^{-1}} \times \frac{5 x^{-3} y^{-1}}{1} }

Step 2: Combine Like Terms

Now, we will combine the like terms in the numerator and denominator. This will give us:

10x2yβˆ’4Γ—5xβˆ’3yβˆ’15xβˆ’3yβˆ’1{ \frac{10 x^2 y^{-4} \times 5 x^{-3} y^{-1}}{5 x^{-3} y^{-1}} }

Simplifying this expression, we get:

50x2xβˆ’3yβˆ’4yβˆ’15xβˆ’3yβˆ’1{ \frac{50 x^2 x^{-3} y^{-4} y^{-1}}{5 x^{-3} y^{-1}} }

Step 3: Simplify the Expression

Finally, we will simplify the resulting expression by canceling out any common factors. This will give us:

50x2βˆ’3yβˆ’4βˆ’15xβˆ’3yβˆ’1{ \frac{50 x^{2-3} y^{-4-1}}{5 x^{-3} y^{-1}} }

Simplifying this expression, we get:

50xβˆ’1yβˆ’55xβˆ’3yβˆ’1{ \frac{50 x^{-1} y^{-5}}{5 x^{-3} y^{-1}} }

Final Answer

The simplified expression with no negative exponents is:

10xyβˆ’4{ 10 x y^{-4} }

This is the final answer.

Conclusion

Simplifying expressions with negative exponents requires a step-by-step approach. By following the steps outlined in this article, we can rewrite the given expression with no negative exponents. This skill is essential in mathematics and is used extensively in algebra, calculus, and other branches of mathematics.

Frequently Asked Questions

Q: What is a negative exponent?

A: A negative exponent indicates that the base is being raised to a power that is the reciprocal of the given exponent.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you can use the following steps:

  1. Invert and multiply
  2. Combine like terms
  3. Simplify the expression

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 10xyβˆ’410 x y^{-4}.

Additional Resources

For more information on simplifying expressions with negative exponents, you can refer to the following resources:

  • Khan Academy: Simplifying Expressions with Negative Exponents
  • Mathway: Simplifying Expressions with Negative Exponents
  • Wolfram Alpha: Simplifying Expressions with Negative Exponents

By following the steps outlined in this article and using the additional resources provided, you can master the skill of simplifying expressions with negative exponents.

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Introduction

In our previous article, we explored how to simplify the given expression by rewriting it with no negative exponents. In this article, we will provide a Q&A guide to help you understand the concept better.

Q&A Guide

Q: What is a negative exponent?

A: A negative exponent indicates that the base is being raised to a power that is the reciprocal of the given exponent. In other words, a negative exponent is equivalent to taking the reciprocal of the base and raising it to the positive exponent.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you can use the following steps:

  1. Invert and Multiply: Invert the second fraction and multiply it by the first fraction.
  2. Combine Like Terms: Combine the like terms in the numerator and denominator.
  3. Simplify the Expression: Simplify the resulting expression by canceling out any common factors.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the base is being raised to a power that is greater than 1. A negative exponent indicates that the base is being raised to a power that is less than 1.

Q: How do I handle negative exponents in the numerator and denominator?

A: When handling negative exponents in the numerator and denominator, you can use the following rules:

  • When a negative exponent is in the numerator, move it to the denominator by changing its sign.
  • When a negative exponent is in the denominator, move it to the numerator by changing its sign.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 10xyβˆ’410 x y^{-4}.

Q: Can I simplify an expression with multiple negative exponents?

A: Yes, you can simplify an expression with multiple negative exponents by following the same steps as before. However, you may need to use the rules for handling negative exponents in the numerator and denominator.

Q: How do I know when to use the rules for handling negative exponents?

A: You should use the rules for handling negative exponents when you have a negative exponent in the numerator or denominator. This will help you simplify the expression correctly.

Examples

Example 1: Simplifying an Expression with a Negative Exponent in the Numerator

Simplify the expression: 2xβˆ’23x3\frac{2 x^{-2}}{3 x^3}

To simplify this expression, we will use the following steps:

  1. Invert and multiply
  2. Combine like terms
  3. Simplify the expression

The final answer is: 23x5\frac{2}{3 x^5}

Example 2: Simplifying an Expression with a Negative Exponent in the Denominator

Simplify the expression: 4x2xβˆ’3\frac{4 x^2}{x^{-3}}

To simplify this expression, we will use the following steps:

  1. Invert and multiply
  2. Combine like terms
  3. Simplify the expression

The final answer is: 4x54 x^5

Conclusion

Simplifying expressions with negative exponents requires a step-by-step approach. By following the steps outlined in this article and using the Q&A guide, you can master the skill of simplifying expressions with negative exponents.

Frequently Asked Questions

Q: What is a negative exponent?

A: A negative exponent indicates that the base is being raised to a power that is the reciprocal of the given exponent.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you can use the following steps:

  1. Invert and multiply
  2. Combine like terms
  3. Simplify the expression

Q: What is the final answer to the given expression?

A: The final answer to the given expression is 10xyβˆ’410 x y^{-4}.

Additional Resources

For more information on simplifying expressions with negative exponents, you can refer to the following resources:

  • Khan Academy: Simplifying Expressions with Negative Exponents
  • Mathway: Simplifying Expressions with Negative Exponents
  • Wolfram Alpha: Simplifying Expressions with Negative Exponents

By following the steps outlined in this article and using the additional resources provided, you can master the skill of simplifying expressions with negative exponents.