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

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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'll explore how to simplify the expression $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$ and determine which of the given options is equivalent.

Understanding Exponents

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

Simplifying the Expression

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To simplify the given expression, we'll start by applying the rules of exponents. We can begin 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 exponents as positive exponents by taking the reciprocal of the base:

$$-9 x^{-1} y^{-9} = -9 \left(\frac{1}{x}\right) \left(\frac{1}{y^9}\right)$$

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:

$$-15 x^5 y^{-3} = -15 x^5 \left(\frac{1}{y^3}\right)$$

Combining the Numerator and Denominator

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

$$\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}} = \frac{-9 \left(\frac{1}{x}\right) \left(\frac{1}{y^9}\right)}{-15 x^5 \left(\frac{1}{y^3}\right)}$$

Canceling Out Common Factors

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We can simplify the expression further by canceling out common factors in the numerator and denominator. We can start by canceling out the negative signs:

$$\frac{-9 \left(\frac{1}{x}\right) \left(\frac{1}{y^9}\right)}{-15 x^5 \left(\frac{1}{y^3}\right)} = \frac{9 \left(\frac{1}{x}\right) \left(\frac{1}{y^9}\right)}{15 x^5 \left(\frac{1}{y^3}\right)}$$

Next, we can cancel out the common factors in the numerator and denominator:

$$\frac{9 \left(\frac{1}{x}\right) \left(\frac{1}{y^9}\right)}{15 x^5 \left(\frac{1}{y^3}\right)} = \frac{9}{15 x^6 y^6}$$

Final Simplification

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We can simplify the expression further by dividing both the numerator and denominator by their greatest common divisor, which is 3:

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

Conclusion

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In conclusion, the simplified expression is $\frac{3}{5 x^6 y^6}$. This expression is equivalent to the original expression $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$. Therefore, the correct answer is:

$\boxed{\frac{3}{5 x^6 y^6}}$

This expression is equivalent to option B, $\frac{3}{5 x^6 y^6}$.

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Introduction

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In our previous article, we explored how to simplify the expression $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$ and determined that the simplified expression is $\frac{3}{5 x^6 y^6}$. In this article, we'll answer some frequently asked questions about simplifying exponential expressions.

Q&A

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Q: What is the rule for simplifying negative exponents?

A: When dealing with negative exponents, we can rewrite them as positive exponents by taking the reciprocal of the base. For instance, $x^{-3}$ is equivalent to $\frac{1}{x^3}$.

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

A: To simplify an expression with multiple negative exponents, we can start by rewriting each negative exponent as a positive exponent by taking the reciprocal of the base. Then, we can combine the terms using the rules of exponents.

Q: What is the difference between $x^{-3}$ and $\frac{1}{x^3}$?

A: $x^{-3}$ and $\frac{1}{x^3}$ are equivalent expressions. The negative exponent $x^{-3}$ means $\frac{1}{x^3}$.

Q: How do I simplify an expression with a negative exponent in the denominator?

A: To simplify an expression with a negative exponent in the denominator, we can start by rewriting the negative exponent as a positive exponent by taking the reciprocal of the base. Then, we can combine the terms using the rules of exponents.

Q: What is the rule for canceling out common factors in the numerator and denominator?

A: When simplifying an expression, we can cancel out common factors in the numerator and denominator. This involves dividing both the numerator and denominator by their greatest common divisor.

Q: How do I determine the greatest common divisor of two numbers?

A: To determine the greatest common divisor of two numbers, we can list the factors of each number and find the largest factor that they have in common.

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

A: The final simplified expression for $\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}$ is $\frac{3}{5 x^6 y^6}$.

Conclusion

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In conclusion, simplifying exponential expressions involves understanding the rules of exponents and applying them to rewrite negative exponents as positive exponents. We can then combine the terms using the rules of exponents and cancel out common factors in the numerator and denominator. By following these steps, we can simplify complex expressions and determine their equivalent forms.

Common Mistakes to Avoid

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• Failing to rewrite negative exponents as positive exponents
• Not combining terms using the rules of exponents
• Not canceling out common factors in the numerator and denominator
• Not determining the greatest common divisor of two numbers

Tips for Simplifying Exponential Expressions

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• Start by rewriting negative exponents as positive exponents
• Combine terms using the rules of exponents
• Cancel out common factors in the numerator and denominator
• Determine the greatest common divisor of two numbers
• Check your work by plugging in values for the variables

Practice Problems

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• Simplify the expression $\frac{2 x^{-2} y^3}{3 x^4 y^{-2}}$
• Simplify the expression $\frac{-4 x^3 y^{-5}}{-2 x^2 y^{-3}}$
• Simplify the expression $\frac{6 x^{-1} y^2}{9 x^3 y^{-1}}$

Solutions

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• $\frac{2 x^{-2} y^3}{3 x^4 y^{-2}} = \frac{2}{3 x^6 y^5}$
• $\frac{-4 x^3 y^{-5}}{-2 x^2 y^{-3}} = \frac{2 x y^{-2}}{1} = 2 x y^2$
• $\frac{6 x^{-1} y^2}{9 x^3 y^{-1}} = \frac{2 x^{-2} y^3}{3 x^3 y^2} = \frac{2}{3 x^5 y^2}$