Simplify The Expression:$\[ \frac{4 X^{-3} Y^4}{8 X^2\left(y^{-3}\right)^2} \\]

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying the given expression: ${ \frac{4 x^{-3} y^4}{8 x^2\left(y{-3}\right)^2} }$. We will break down the process into manageable steps, explaining each step in detail to ensure a thorough understanding of the concept.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at it. The given expression is a fraction, with the numerator being $4 x^{-3} y^4$ and the denominator being $8 x^2\left(y^{-3}\right)^2$. To simplify this expression, we need to apply the rules of exponents and fractions.

Simplifying the Numerator

The numerator of the expression is $4 x^{-3} y^4$. To simplify this, we can start by breaking down the terms. The term $x^{-3}$ can be rewritten as $\frac{1}{x^3}$, and the term $y^4$ remains the same.

Simplifying the Denominator

The denominator of the expression is $8 x^2\left(y^{-3}\right)^2$. To simplify this, we need to apply the rule of exponents, which states that when we raise a power to another power, we multiply the exponents. In this case, we have $\left(y^{-3}\right)^2$, which can be rewritten as $y^{-6}$.

Applying the Rules of Exponents

Now that we have simplified the numerator and denominator, we can apply the rules of exponents to simplify the expression further. When we divide two terms with the same base, we subtract the exponents. In this case, we have $x^{-3}$ in the numerator and $x^2$ in the denominator, so we can subtract the exponents to get $x^{-3-2} = x^{-5}$.

Simplifying the Expression

Now that we have applied the rules of exponents, we can simplify the expression further. We can rewrite the expression as $\frac{4 y^4}{8 x^5 y^6}$. To simplify this, we can cancel out common factors in the numerator and denominator. We can cancel out a factor of 4 in the numerator and denominator, and we can also cancel out a factor of $y^4$ in the numerator and denominator.

Final Simplification

After canceling out common factors, we are left with $\frac{1}{2 x^5 y^2}$. This is the final simplified form of the expression.

Conclusion

Simplifying expressions is an essential skill in mathematics, and it requires a thorough understanding of the rules of exponents and fractions. In this article, we have broken down the process of simplifying the given expression into manageable steps, explaining each step in detail to ensure a thorough understanding of the concept. By following these steps, you can simplify any expression and arrive at the final answer.

Frequently Asked Questions

• What is the rule of exponents? The rule of exponents states that when we raise a power to another power, we multiply the exponents.
• How do we simplify a fraction? To simplify a fraction, we can cancel out common factors in the numerator and denominator.
• What is the final simplified form of the expression? The final simplified form of the expression is $\frac{1}{2 x^5 y^2}$.

Additional Resources

• Khan Academy: Exponents and Fractions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Exponents and Fractions

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires a thorough understanding of the rules of exponents and fractions. By following the steps outlined in this article, you can simplify any expression and arrive at the final answer. Remember to always break down the expression into manageable steps, and to apply the rules of exponents and fractions to simplify the expression further. With practice and patience, you can become proficient in simplifying expressions and arrive at the final answer with confidence.

Introduction

In our previous article, we explored the process of simplifying the expression ${ \frac{4 x^{-3} y^4}{8 x^2\left(y{-3}\right)^2} }$. We broke down the process into manageable steps, explaining each step in detail to ensure a thorough understanding of the concept. In this article, we will continue to explore the topic of simplifying expressions, this time in the form of a Q&A guide.

Q&A: Simplifying Expressions

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to break it down into its individual components. This involves identifying the numerator and denominator, and then simplifying each component separately.

Q: How do I simplify a fraction with negative exponents?

A: To simplify a fraction with negative exponents, you can rewrite the negative exponent as a positive exponent in the denominator. For example, $x^{-3}$ can be rewritten as $\frac{1}{x^3}$.

Q: What is the rule of exponents?

A: The rule of exponents states that when we raise a power to another power, we multiply the exponents. For example, $(x^2)^3 = x^{2 \cdot 3} = x^6$.

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

A: To simplify an expression with multiple variables, you can apply the rules of exponents and fractions separately to each variable. For example, $\frac{4 x^{-3} y^4}{8 x^2\left(y^{-3}\right)^2}$ can be simplified by applying the rules of exponents and fractions to each variable separately.

Q: What is the final simplified form of the expression ${

\frac{4 x^{-3} y^4}{8 x^2\left(y{-3}\right)^2} }$?

A: The final simplified form of the expression is $\frac{1}{2 x^5 y^2}$.

Q: How do I check my work when simplifying an expression?

A: To check your work when simplifying an expression, you can plug in values for the variables and see if the expression simplifies to the expected value. For example, if you plug in $x = 2$ and $y = 3$ into the expression $\frac{4 x^{-3} y^4}{8 x^2\left(y^{-3}\right)^2}$, you should get the expected value of $\frac{1}{2 \cdot 2^5 \cdot 3^2}$.

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

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

• Forgetting to apply the rules of exponents and fractions
• Not simplifying the numerator and denominator separately
• Not checking your work when simplifying an expression

Conclusion

Simplifying expressions is an essential skill in mathematics, and it requires a thorough understanding of the rules of exponents and fractions. By following the steps outlined in this article, you can simplify any expression and arrive at the final answer. Remember to always break down the expression into manageable steps, and to apply the rules of exponents and fractions to simplify the expression further. With practice and patience, you can become proficient in simplifying expressions and arrive at the final answer with confidence.

Frequently Asked Questions

• What is the rule of exponents? The rule of exponents states that when we raise a power to another power, we multiply the exponents.
• How do I simplify a fraction with negative exponents? To simplify a fraction with negative exponents, you can rewrite the negative exponent as a positive exponent in the denominator.
• What is the final simplified form of the expression ${ \frac{4 x^{-3} y^4}{8 x^2\left(y{-3}\right)^2} }$? The final simplified form of the expression is $\frac{1}{2 x^5 y^2}$.

Additional Resources

• Khan Academy: Exponents and Fractions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Exponents and Fractions

Final Thoughts

Simplifying expressions is an essential skill in mathematics, and it requires a thorough understanding of the rules of exponents and fractions. By following the steps outlined in this article, you can simplify any expression and arrive at the final answer. Remember to always break down the expression into manageable steps, and to apply the rules of exponents and fractions to simplify the expression further. With practice and patience, you can become proficient in simplifying expressions and arrive at the final answer with confidence.