Simplify The Expression.$\[ \frac{x^{-3} Y^2}{x^4 Y^{10}} \\]Result:$\[ \frac{1}{x^{[?]} Y} \\]

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Introduction

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Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying a given expression using the properties of exponents. We will break down the process into manageable steps and provide a clear explanation of each step.

The Given Expression

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The given expression is $\frac{x^{-3} y^2}{x^4 y^{10}}$. Our goal is to simplify this expression to the form $\frac{1}{x^{[?]} y}$.

Step 1: Apply the Quotient Rule for Exponents

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To simplify the expression, we will apply the quotient rule for exponents, which states that when dividing two powers with the same base, we subtract the exponents. In this case, we have:

$$\frac{x^{-3} y^2}{x^4 y^{10}} = x^{-3-4} y^{2-10}$$

Step 2: Simplify the Exponents

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Now, we will simplify the exponents by applying the rules of exponents. We have:

$$x^{-3-4} y^{2-10} = x^{-7} y^{-8}$$

Step 3: Rewrite the Expression with Positive Exponents

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To rewrite the expression with positive exponents, we will apply the rule that states $a^{-n} = \frac{1}{a^n}$. In this case, we have:

$$x^{-7} y^{-8} = \frac{1}{x^7 y^8}$$

Step 4: Simplify the Expression Further

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We can simplify the expression further by combining the terms in the denominator. We have:

$$\frac{1}{x^7 y^8} = \frac{1}{x^7} \cdot \frac{1}{y^8}$$

Step 5: Write the Final Answer

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The final answer is $\boxed{\frac{1}{x^7 y^8}}$.

Conclusion

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In this article, we simplified the given expression using the properties of exponents. We applied the quotient rule for exponents, simplified the exponents, and rewrote the expression with positive exponents. Finally, we simplified the expression further by combining the terms in the denominator. The final answer is $\boxed{\frac{1}{x^7 y^8}}$.

Frequently Asked Questions

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Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when dividing two powers with the same base, we subtract the exponents.

Q: How do I simplify exponents?

A: To simplify exponents, you can apply the rules of exponents, such as $a^{-n} = \frac{1}{a^n}$.

Q: How do I rewrite an expression with positive exponents?

A: To rewrite an expression with positive exponents, you can apply the rule that states $a^{-n} = \frac{1}{a^n}$.

Tips and Tricks

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Tip 1: Always apply the quotient rule for exponents when dividing two powers with the same base.

Tip 2: Simplify exponents by applying the rules of exponents.

Tip 3: Rewrite expressions with positive exponents by applying the rule that states $a^{-n} = \frac{1}{a^n}$.

Related Topics

---

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics. In this article, we will focus on simplifying a given expression using the properties of exponents.

Properties of Exponents

The properties of exponents are a set of rules that govern the behavior of exponents. In this article, we will focus on the quotient rule for exponents.

Algebraic Manipulation

Algebraic manipulation is a set of techniques used to simplify and manipulate algebraic expressions. In this article, we will focus on simplifying a given expression using the properties of exponents.

References

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• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for informational purposes only and are not necessarily related to the content of this article.

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Introduction

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In our previous article, we simplified the expression $\frac{x^{-3} y^2}{x^4 y^{10}}$ to the form $\frac{1}{x^7 y^8}$. However, we received many questions from readers who were struggling to understand the process. In this article, we will provide a Q&A guide to help readers who are having trouble simplifying the expression.

Q&A Guide

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Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when dividing two powers with the same base, we subtract the exponents. In other words, if we have $\frac{a^m}{a^n}$, we can simplify it to $a^{m-n}$.

Q: How do I simplify exponents?

A: To simplify exponents, you can apply the rules of exponents, such as $a^{-n} = \frac{1}{a^n}$. You can also use the quotient rule for exponents to simplify expressions.

Q: How do I rewrite an expression with positive exponents?

A: To rewrite an expression with positive exponents, you can apply the rule that states $a^{-n} = \frac{1}{a^n}$. For example, if we have $x^{-3}$, we can rewrite it as $\frac{1}{x^3}$.

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

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a power and then taken as a reciprocal.

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

A: To simplify an expression with multiple bases, you can apply the rules of exponents separately to each base. For example, if we have $\frac{x^2 y^3}{x^4 y^5}$, we can simplify it to $\frac{x^{2-4} y^{3-5}}{1}$.

Q: What is the final answer to the expression $\frac{x^{-3} y^2}{x^4 y^{10}}$?

A: The final answer to the expression $\frac{x^{-3} y^2}{x^4 y^{10}}$ is $\frac{1}{x^7 y^8}$.

Tips and Tricks

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Tip 1: Always apply the quotient rule for exponents when dividing two powers with the same base.

Tip 2: Simplify exponents by applying the rules of exponents.

Tip 3: Rewrite expressions with positive exponents by applying the rule that states $a^{-n} = \frac{1}{a^n}$.

Common Mistakes

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Mistake 1: Not applying the quotient rule for exponents when dividing two powers with the same base.

Mistake 2: Not simplifying exponents by applying the rules of exponents.

Mistake 3: Not rewriting expressions with positive exponents by applying the rule that states $a^{-n} = \frac{1}{a^n}$.

Conclusion

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In this article, we provided a Q&A guide to help readers who are having trouble simplifying the expression $\frac{x^{-3} y^2}{x^4 y^{10}}$. We covered topics such as the quotient rule for exponents, simplifying exponents, and rewriting expressions with positive exponents. We also provided tips and tricks to help readers avoid common mistakes.

Frequently Asked Questions

---

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when dividing two powers with the same base, we subtract the exponents.

Q: How do I simplify exponents?

A: To simplify exponents, you can apply the rules of exponents, such as $a^{-n} = \frac{1}{a^n}$.

Q: How do I rewrite an expression with positive exponents?

A: To rewrite an expression with positive exponents, you can apply the rule that states $a^{-n} = \frac{1}{a^n}$.

Related Topics

---

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics. In this article, we will focus on simplifying a given expression using the properties of exponents.

Properties of Exponents

The properties of exponents are a set of rules that govern the behavior of exponents. In this article, we will focus on the quotient rule for exponents.

Algebraic Manipulation

Algebraic manipulation is a set of techniques used to simplify and manipulate algebraic expressions. In this article, we will focus on simplifying a given expression using the properties of exponents.

References

---

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for informational purposes only and are not necessarily related to the content of this article.