Which Expression Is Equivalent To − 9 X − 1 Y − 9 − 15 X 5 Y − 3 \frac{-9 X^{-1} Y^{-9}}{-15 X^5 Y^{-3}} − 15 X 5 Y − 3 − 9 X − 1 Y − 9 ​ ? Assume X ≠ 0 , Y ≠ 0 X \neq 0, Y \neq 0 X  = 0 , Y  = 0 .A. 3 5 X 5 Y 3 \frac{3}{5 X^5 Y^3} 5 X 5 Y 3 3 ​ B. 3 5 X 6 Y 6 \frac{3}{5 X^6 Y^6} 5 X 6 Y 6 3 ​ C. 5 3 X 5 Y 3 \frac{5}{3 X^5 Y^3} 3 X 5 Y 3 5 ​ D. $\frac{5}{3 X^6

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Introduction

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When dealing with exponential expressions, it's essential to understand the rules of exponents to simplify complex expressions. In this article, we will focus on simplifying the expression $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$ and determine which of the given options is equivalent to it.

Understanding Exponents

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Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ can be written as $x \cdot x \cdot x$. When dealing with negative exponents, we can rewrite them as positive exponents by taking the reciprocal of the base. For instance, $x^{-3}$ can be written as $\frac{1}{x^3}$.

Simplifying the Expression

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To simplify the given expression, we need to apply the rules of exponents. We can start by simplifying the numerator and denominator separately.

Simplifying the Numerator

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The numerator is $-9 x^{-1} y^{-9}$. We can rewrite the negative exponent as a positive exponent by taking the reciprocal of the base:

$$-9 x^{-1} y^{-9} = -9 \cdot \frac{1}{x} \cdot \frac{1}{y^9}$$

Simplifying the Denominator

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The denominator is $-15 x^5 y^{-3}$. We can rewrite the negative exponent as a positive exponent by taking the reciprocal of the base:

$$-15 x^5 y^{-3} = -15 \cdot x^5 \cdot \frac{1}{y^3}$$

Combining the Numerator and Denominator

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

$$\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}} = \frac{-9 \cdot \frac{1}{x} \cdot \frac{1}{y^9}}{-15 \cdot x^5 \cdot \frac{1}{y^3}}$$

Canceling Out Common Factors

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We can cancel out common factors in the numerator and denominator to simplify the expression further:

$$\frac{-9 \cdot \frac{1}{x} \cdot \frac{1}{y^9}}{-15 \cdot x^5 \cdot \frac{1}{y^3}} = \frac{-9 \cdot y^3}{-15 \cdot x^5 \cdot x}$$

Simplifying the Expression Further

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

$$\frac{-9 \cdot y^3}{-15 \cdot x^5 \cdot x} = \frac{3 \cdot y^3}{5 \cdot x^6}$$

Conclusion

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The simplified expression is $\frac{3 \cdot y^3}{5 \cdot x^6}$. We can rewrite this expression as $\frac{3}{5 x^6 y^3}$.

Comparison with Options

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

• Option A: $\frac{3}{5 x^5 y^3}$
• Option B: $\frac{3}{5 x^6 y^6}$
• Option C: $\frac{5}{3 x^5 y^3}$
• Option D: $\frac{5}{3 x^6 y^6}$

Determining the Equivalent Expression

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Based on the simplified expression, we can determine that the equivalent expression is:

• Option A: $\frac{3}{5 x^5 y^3}$

This is because the simplified expression $\frac{3 \cdot y^3}{5 \cdot x^6}$ is equivalent to $\frac{3}{5 x^5 y^3}$.

Final Answer

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The final answer is:

• Option A: $\frac{3}{5 x^5 y^3}$

This is the equivalent expression to the given expression $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$.

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Introduction

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In our previous article, we discussed how to simplify the expression $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$. We also determined that the equivalent expression is $\frac{3}{5 x^5 y^3}$. In this article, we will answer some frequently asked questions related to simplifying exponential expressions.

Q&A

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Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions is to apply the following steps:

1. Simplify the numerator and denominator separately.
2. Cancel out common factors in the numerator and denominator.
3. Simplify the expression further by applying the rules of exponents.

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

A: To simplify an expression with negative exponents, you can rewrite the negative exponent as a positive exponent by taking the reciprocal of the base. For example, $x^{-3}$ can be written as $\frac{1}{x^3}$.

Q: What is the difference between $x^3$ and $x^{-3}$?

A: $x^3$ and $x^{-3}$ are not the same. $x^3$ represents $x$ multiplied by itself three times, while $x^{-3}$ represents the reciprocal of $x^3$.

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

A: To simplify an expression with multiple variables, you can apply the same rules as before. Simplify the numerator and denominator separately, cancel out common factors, and simplify the expression further by applying the rules of exponents.

Q: What is the equivalent expression to $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$?

A: The equivalent expression to $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$ is $\frac{3}{5 x^5 y^3}$.

Q: How do I determine the equivalent expression to a given expression?

A: To determine the equivalent expression to a given expression, you can simplify the expression by applying the rules of exponents and canceling out common factors.

Conclusion

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Simplifying exponential expressions can be a challenging task, but by applying the rules of exponents and canceling out common factors, you can simplify complex expressions and determine the equivalent expression. We hope this article has helped you understand how to simplify exponential expressions and answer some frequently asked questions.

Additional Resources

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If you need additional help or resources, you can check out the following:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponential Expressions
• Wolfram Alpha: Exponential Expressions

Final Answer

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The final answer is:

• Option A: $\frac{3}{5 x^5 y^3}$

This is the equivalent expression to the given expression $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$.