Simplifying Rational ExpressionsWhich Shows The Following Expression After The Negative Exponents Have Been Eliminated? X Y − 6 X − 4 Y 2 , X ≠ 0 , Y ≠ 0 \frac{x Y^{-6}}{x^{-4} Y^2}, X \neq 0, Y \neq 0 X − 4 Y 2 X Y − 6 ​ , X  = 0 , Y  = 0 A. X X 4 Y 2 Y 6 \frac{x X^4}{y^2 Y^6} Y 2 Y 6 X X 4 ​ B. X 4 Y 2 X Y 6 \frac{x^4 Y^2}{x Y^6} X Y 6 X 4 Y 2 ​ C.

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Introduction

Simplifying rational expressions is a crucial skill in algebra, as it allows us to rewrite complex expressions in a more manageable form. In this article, we will focus on simplifying rational expressions with negative exponents. We will use the given expression xy−6x−4y2,x≠0,y≠0\frac{x y^{-6}}{x^{-4} y^2}, x \neq 0, y \neq 0 as an example and show how to eliminate the negative exponents.

Understanding Negative Exponents

Before we dive into simplifying the given expression, let's take a moment to understand what negative exponents mean. A negative exponent is a shorthand way of writing a fraction with a positive exponent. For example, x−nx^{-n} is equivalent to 1xn\frac{1}{x^n}. This means that when we see a negative exponent, we can rewrite it as a fraction with a positive exponent.

Simplifying the Given Expression

Now that we understand negative exponents, let's simplify the given expression xy−6x−4y2,x≠0,y≠0\frac{x y^{-6}}{x^{-4} y^2}, x \neq 0, y \neq 0. To simplify this expression, we need to eliminate the negative exponents. We can do this by rewriting the negative exponents as fractions with positive exponents.

Step 1: Rewrite Negative Exponents as Fractions

We start by rewriting the negative exponents as fractions with positive exponents. We have y−6y^{-6} and x−4x^{-4}. We can rewrite these as 1y6\frac{1}{y^6} and 1x4\frac{1}{x^4}, respectively.

Step 2: Rewrite the Expression with Positive Exponents

Now that we have rewritten the negative exponents as fractions with positive exponents, we can rewrite the entire expression with positive exponents. We have:

x1y61x4y2\frac{x \frac{1}{y^6}}{\frac{1}{x^4} y^2}

Step 3: Simplify the Expression

Now that we have rewritten the expression with positive exponents, we can simplify it. We can start by canceling out any common factors in the numerator and denominator. In this case, we have xx in the numerator and 1x4\frac{1}{x^4} in the denominator. We can cancel out the xx by multiplying the numerator and denominator by x3x^3. This gives us:

x1y6x31x4y2x3\frac{x \frac{1}{y^6} x^3}{\frac{1}{x^4} y^2 x^3}

Step 4: Simplify the Expression Further

Now that we have canceled out the xx, we can simplify the expression further. We can start by combining the fractions in the numerator and denominator. We have 1y6\frac{1}{y^6} in the numerator and 1x4\frac{1}{x^4} in the denominator. We can combine these fractions by multiplying them together. This gives us:

xx3y6y2\frac{x x^3}{y^6 y^2}

Step 5: Simplify the Expression Even Further

Now that we have combined the fractions, we can simplify the expression even further. We can start by combining the exponents in the numerator and denominator. We have x3x^3 in the numerator and y6y2y^6 y^2 in the denominator. We can combine these exponents by adding them together. This gives us:

x1+3y6+2\frac{x^{1+3}}{y^{6+2}}

Step 6: Simplify the Expression to Its Final Form

Now that we have combined the exponents, we can simplify the expression to its final form. We have x4y8\frac{x^4}{y^8}. This is the simplified form of the given expression.

Conclusion

In this article, we have shown how to simplify a rational expression with negative exponents. We started by rewriting the negative exponents as fractions with positive exponents, and then we simplified the expression by canceling out common factors and combining the fractions. We ended up with the simplified form of the expression, which is x4y8\frac{x^4}{y^8}.

Final Answer

The final answer is x4y8\boxed{\frac{x^4}{y^8}}.

Discussion

The given expression xy−6x−4y2,x≠0,y≠0\frac{x y^{-6}}{x^{-4} y^2}, x \neq 0, y \neq 0 can be simplified to x4y2xy6\frac{x^4 y^2}{x y^6}. This is the correct answer.

Comparison of Options

Let's compare the options:

  • Option A: xx4y2y6\frac{x x^4}{y^2 y^6}
  • Option B: x4y2xy6\frac{x^4 y^2}{x y^6}
  • Option C: (no option)

Option B is the correct answer.

Final Thoughts

Simplifying rational expressions with negative exponents requires a good understanding of negative exponents and how to rewrite them as fractions with positive exponents. By following the steps outlined in this article, you can simplify any rational expression with negative exponents.

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Introduction

In our previous article, we discussed how to simplify rational expressions with negative exponents. We used the given expression xy−6x−4y2,x≠0,y≠0\frac{x y^{-6}}{x^{-4} y^2}, x \neq 0, y \neq 0 as an example and showed how to eliminate the negative exponents. In this article, we will answer some frequently asked questions about simplifying rational expressions with negative exponents.

Q&A

Q: What is the first step in simplifying a rational expression with negative exponents?

A: The first step in simplifying a rational expression with negative exponents is to rewrite the negative exponents as fractions with positive exponents.

Q: How do I rewrite a negative exponent as a fraction with a positive exponent?

A: To rewrite a negative exponent as a fraction with a positive exponent, you can use the following rule: x−n=1xnx^{-n} = \frac{1}{x^n}.

Q: What is the next step in simplifying a rational expression with negative exponents?

A: After rewriting the negative exponents as fractions with positive exponents, the next step is to simplify the expression by canceling out any common factors in the numerator and denominator.

Q: How do I simplify a rational expression with negative exponents?

A: To simplify a rational expression with negative exponents, you can follow these steps:

  1. Rewrite the negative exponents as fractions with positive exponents.
  2. Simplify the expression by canceling out any common factors in the numerator and denominator.
  3. Combine the fractions in the numerator and denominator.
  4. Simplify the expression further by combining the exponents in the numerator and denominator.

Q: What is the final step in simplifying a rational expression with negative exponents?

A: The final step in simplifying a rational expression with negative exponents is to simplify the expression to its final form.

Q: How do I know if I have simplified a rational expression with negative exponents correctly?

A: To check if you have simplified a rational expression with negative exponents correctly, you can plug in some values for the variables and see if the expression simplifies to the expected value.

Q: What are some common mistakes to avoid when simplifying rational expressions with negative exponents?

A: Some common mistakes to avoid when simplifying rational expressions with negative exponents include:

  • Not rewriting the negative exponents as fractions with positive exponents.
  • Not canceling out common factors in the numerator and denominator.
  • Not combining the fractions in the numerator and denominator.
  • Not simplifying the expression further by combining the exponents in the numerator and denominator.

Conclusion

Simplifying rational expressions with negative exponents requires a good understanding of negative exponents and how to rewrite them as fractions with positive exponents. By following the steps outlined in this article, you can simplify any rational expression with negative exponents. Remember to avoid common mistakes and check your work to ensure that you have simplified the expression correctly.

Final Thoughts

Simplifying rational expressions with negative exponents is an important skill in algebra, and it requires practice to become proficient. By working through examples and practicing simplifying rational expressions with negative exponents, you can become more confident and proficient in this area.

Common Mistakes to Avoid

  • Not rewriting the negative exponents as fractions with positive exponents.
  • Not canceling out common factors in the numerator and denominator.
  • Not combining the fractions in the numerator and denominator.
  • Not simplifying the expression further by combining the exponents in the numerator and denominator.

Tips for Simplifying Rational Expressions with Negative Exponents

  • Make sure to rewrite the negative exponents as fractions with positive exponents.
  • Simplify the expression by canceling out any common factors in the numerator and denominator.
  • Combine the fractions in the numerator and denominator.
  • Simplify the expression further by combining the exponents in the numerator and denominator.
  • Check your work to ensure that you have simplified the expression correctly.

Practice Problems

  1. Simplify the rational expression xy−3x−2y4,x≠0,y≠0\frac{x y^{-3}}{x^{-2} y^4}, x \neq 0, y \neq 0.
  2. Simplify the rational expression xz−2x−1z3,x≠0,z≠0\frac{x z^{-2}}{x^{-1} z^3}, x \neq 0, z \neq 0.
  3. Simplify the rational expression y−1z2y3z−4,y≠0,z≠0\frac{y^{-1} z^2}{y^3 z^{-4}}, y \neq 0, z \neq 0.

Solutions

  1. x3y4yz2\frac{x^3 y^4}{y z^2}
  2. x2z5z2\frac{x^2 z^5}{z^2}
  3. z6y4z2\frac{z^6}{y^4 z^2}