Simplify The Expression:$\[ \frac{\left(x^3 Y\right)^2}{x Y^8} = \\]

Feb 28, 2025 by ADMIN 70 views

**Simplify the Expression: A Comprehensive Guide**

Introduction

In this article, we will delve into the world of algebra and simplify a given expression involving exponents and variables. The expression we will be working with is $\frac{\left(x^3 y\right)^2}{x y^8}$ . Our goal is to simplify this expression and provide a clear understanding of the steps involved.

Understanding Exponents

Before we begin simplifying the expression, it's essential to understand the concept of exponents. An exponent is a small number that is written to the upper right of a number or a variable. It represents the power to which the base is raised. For example, in the expression $x^3$ , the exponent 3 represents the power to which the base $x$ is raised.

Simplifying the Expression

To simplify the expression $\frac{\left(x^3 y\right)^2}{x y^8}$ , we will use the properties of exponents. The first step is to expand the numerator using the power rule of exponents, which states that $\left(a^m\right)^n = a^{mn}$ .

\left(x^3 y\right)^2 = x^{3 \cdot 2} y^2 = x^6 y^2

Now that we have expanded the numerator, we can rewrite the original expression as $\frac{x^6 y^2}{x y^8}$ .

Applying the Quotient Rule

The next step is to apply the quotient rule of exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$ . We can apply this rule to the expression $\frac{x^6 y^2}{x y^8}$ by subtracting the exponents of the variables in the numerator and denominator.

\frac{x^6 y^2}{x y^8} = x^{6-1} y^{2-8} = x^5 y^{-6}

Simplifying Negative Exponents

Now that we have applied the quotient rule, we are left with a negative exponent in the expression $x^5 y^{-6}$ . To simplify this expression, we can use the property of negative exponents, which states that $a^{-m} = \frac{1}{a^m}$ .

x^5 y^{-6} = \frac{x^5}{y^6}

Conclusion

In this article, we have simplified the expression $\frac{\left(x^3 y\right)^2}{x y^8}$ using the properties of exponents. We expanded the numerator using the power rule of exponents, applied the quotient rule to simplify the expression, and finally simplified the negative exponent using the property of negative exponents. The simplified expression is $\frac{x^5}{y^6}$ .

Final Answer

The final answer is $\boxed{\frac{x^5}{y^6}}$ .

Common Mistakes to Avoid

When simplifying expressions involving exponents, it's essential to avoid common mistakes. Here are a few mistakes to watch out for:

Not expanding the numerator: Failing to expand the numerator using the power rule of exponents can lead to incorrect simplification.
Not applying the quotient rule: Failing to apply the quotient rule can result in incorrect simplification.
Not simplifying negative exponents: Failing to simplify negative exponents using the property of negative exponents can lead to incorrect simplification.

Tips and Tricks

Here are a few tips and tricks to help you simplify expressions involving exponents:

Use the power rule of exponents: The power rule of exponents states that $\left(a^m\right)^n = a^{mn}$ . Use this rule to expand the numerator.
Apply the quotient rule: The quotient rule states that $\frac{a^m}{a^n} = a^{m-n}$ . Use this rule to simplify the expression.
Simplify negative exponents: The property of negative exponents states that $a^{-m} = \frac{1}{a^m}$ . Use this property to simplify negative exponents.

Practice Problems

Here are a few practice problems to help you practice simplifying expressions involving exponents:

Simplify the expression $\frac{\left(x^2 y\right)^3}{x y^5}$ .
Simplify the expression $\frac{x^4 y^{-2}}{x^{-1} y^3}$ .
Simplify the expression $\frac{\left(x^3 y^2\right)^2}{x^2 y^4}$ .

Conclusion

Q&A: Simplifying Expressions Involving Exponents

Q: What is the power rule of exponents?

A: The power rule of exponents states that $\left(a^m\right)^n = a^{mn}$ . This means that when you raise a power to another power, you multiply the exponents.

Q: How do I apply the power rule of exponents?

A: To apply the power rule of exponents, simply multiply the exponents of the base. For example, if you have $\left(x^3 y\right)^2$ , you would multiply the exponents to get $x^{3 \cdot 2} y^2 = x^6 y^2$ .

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that $\frac{a^m}{a^n} = a^{m-n}$ . This means that when you divide two powers with the same base, you subtract the exponents.

Q: How do I apply the quotient rule of exponents?

A: To apply the quotient rule of exponents, simply subtract the exponents of the base. For example, if you have $\frac{x^6 y^2}{x y^8}$ , you would subtract the exponents to get $x^{6-1} y^{2-8} = x^5 y^{-6}$ .

Q: What is the property of negative exponents?

A: The property of negative exponents states that $a^{-m} = \frac{1}{a^m}$ . This means that a negative exponent can be rewritten as a fraction with the base in the denominator.

Q: How do I simplify negative exponents?

A: To simplify negative exponents, simply rewrite the expression as a fraction with the base in the denominator. For example, if you have $x^5 y^{-6}$ , you would rewrite it as $\frac{x^5}{y^6}$ .

Q: What are some common mistakes to avoid when simplifying expressions involving exponents?

A: Some common mistakes to avoid when simplifying expressions involving exponents include:

Not expanding the numerator using the power rule of exponents
Not applying the quotient rule of exponents
Not simplifying negative exponents using the property of negative exponents

Q: What are some tips and tricks for simplifying expressions involving exponents?

A: Some tips and tricks for simplifying expressions involving exponents include:

Using the power rule of exponents to expand the numerator
Applying the quotient rule of exponents to simplify the expression
Simplifying negative exponents using the property of negative exponents

Q: How can I practice simplifying expressions involving exponents?

A: You can practice simplifying expressions involving exponents by working through practice problems. Some examples of practice problems include:

Simplify the expression $\frac{\left(x^2 y\right)^3}{x y^5}$
Simplify the expression $\frac{x^4 y^{-2}}{x^{-1} y^3}$
Simplify the expression $\frac{\left(x^3 y^2\right)^2}{x^2 y^4}$

Conclusion

In this article, we have provided a comprehensive guide to simplifying expressions involving exponents. We have covered the power rule of exponents, the quotient rule of exponents, and the property of negative exponents. We have also provided tips and tricks for simplifying expressions involving exponents and practice problems to help you practice your skills.