Not using the rule for negative exponents

Oct 13, 2025 by talynt.org 116 views

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Introduction

In this article, we will simplify a given expression involving exponents and fractions. The expression is $${ \frac{\frac{(x+y)^{-2}}{y{-3}}}{\frac{x^3 y^{-2}(x+y){-1}}{x^{-2}}} }$$. Our goal is to simplify this expression and choose the correct answer from the given options.

Understanding Exponents and Fractions

Before we dive into simplifying the expression, let's review some key concepts related to exponents and fractions.

Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $x^2$ means $x \times x$ , and $x^3$ means $x \times x \times x$ . When we have a negative exponent, it means we are taking the reciprocal of the expression. For example, $x^{-2}$ means $\frac{1}{x^2}$ .

Fractions

Fractions are a way of representing part of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). When we divide one fraction by another, we can multiply the first fraction by the reciprocal of the second fraction.

Simplifying the Expression

Now that we have reviewed the basics of exponents and fractions, let's simplify the given expression.

Step 1: Simplify the Numerator

The numerator of the expression is $\frac{(x+y)^{-2}}{y^{-3}}$ . We can simplify this by using the rule for negative exponents: $x^{-n} = \frac{1}{x^n}$ . Applying this rule, we get:

\frac{(x+y)^{-2}}{y^{-3}} = \frac{1}{(x+y)^2} \times y^3

Step 2: Simplify the Denominator

The denominator of the expression is $\frac{x^3 y^{-2}(x+y)^{-1}}{x^{-2}}$ . We can simplify this by using the rule for negative exponents: $x^{-n} = \frac{1}{x^n}$ . Applying this rule, we get:

\frac{x^3 y^{-2}(x+y)^{-1}}{x^{-2}} = x^5 y^{-2} (x+y)^{-1} \times x^2

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator:

\frac{\frac{1}{(x+y)^2} \times y^3}{x^5 y^{-2} (x+y)^{-1} \times x^2}

We can simplify this expression by canceling out common factors:

\frac{y^3}{x^5 (x+y)^2} \times \frac{(x+y)^2}{x^2 y^{-2}}

This simplifies to:

\frac{y^5}{x^2 + x^5 y^{-2}}

Choosing the Correct Answer

Now that we have simplified the expression, let's compare it to the given options:

A. $\frac{y^5}{x^6 + x^5 y}$ B. $\frac{x+y}{xy}$ C. $\frac{y^5}{x^2 + x^5 y^{-2}}$

The correct answer is C. $\frac{y^5}{x^2 + x^5 y^{-2}}$ , which is the simplified expression we obtained earlier.

Conclusion

In this article, we simplified a given expression involving exponents and fractions. We reviewed the basics of exponents and fractions, and then applied these concepts to simplify the expression. The correct answer is C. $\frac{y^5}{x^2 + x^5 y^{-2}}$ , which is the simplified expression we obtained earlier.

Final Answer

The final answer is C. $\frac{y^5}{x^2 + x^5 y^{-2}}$ .

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Introduction

In our previous article, we simplified a given expression involving exponents and fractions. In this article, we will provide a Q&A guide to help you understand the concepts and steps involved in simplifying the expression.

Q&A Guide

Q: What is the rule for negative exponents?

A: The rule for negative exponents is $x^{-n} = \frac{1}{x^n}$ . This means that when we have a negative exponent, we can take the reciprocal of the expression.

Q: How do we simplify the numerator of the expression?

A: To simplify the numerator, we can use the rule for negative exponents. We can rewrite the numerator as $\frac{1}{(x+y)^2} \times y^3$ .

Q: How do we simplify the denominator of the expression?

A: To simplify the denominator, we can use the rule for negative exponents. We can rewrite the denominator as $x^5 y^{-2} (x+y)^{-1} \times x^2$ .

Q: How do we simplify the expression?

A: To simplify the expression, we can divide the numerator by the denominator. We can simplify the expression by canceling out common factors.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{y^5}{x^2 + x^5 y^{-2}}$ .

Q: How do we choose the correct answer?

A: To choose the correct answer, we can compare the simplified expression to the given options. The correct answer is C. $\frac{y^5}{x^2 + x^5 y^{-2}}$ .

Common Mistakes to Avoid

When simplifying the expression, there are several common mistakes to avoid:

Not using the rule for negative exponents: Make sure to use the rule for negative exponents when simplifying the numerator and denominator.
Not canceling out common factors: Make sure to cancel out common factors when simplifying the expression.
Not comparing the simplified expression to the given options: Make sure to compare the simplified expression to the given options to choose the correct answer.

Tips and Tricks

Here are some tips and tricks to help you simplify the expression:

Use the rule for negative exponents: Make sure to use the rule for negative exponents when simplifying the numerator and denominator.
Cancel out common factors: Make sure to cancel out common factors when simplifying the expression.
Compare the simplified expression to the given options: Make sure to compare the simplified expression to the given options to choose the correct answer.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and steps involved in simplifying the expression. We also discussed common mistakes to avoid and provided tips and tricks to help you simplify the expression.

Final Answer

The final answer is C. $\frac{y^5}{x^2 + x^5 y^{-2}}$ .