Simplify 12 Y 7 18 Y − 3 \frac{12 Y^7}{18 Y^{-3}} 18 Y − 3 12 Y 7 ​ Assuming Y ≠ 0 Y \neq 0 Y  = 0 .A. Y 10 6 \frac{y^{10}}{6} 6 Y 10 ​ B. 2 3 Y 10 \frac{2}{3 Y^{10}} 3 Y 10 2 ​ C. 2 Y 10 3 \frac{2 Y^{10}}{3} 3 2 Y 10 ​

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying a specific algebraic expression, $\frac{12 y^7}{18 y^{-3}}$, assuming $y \neq 0$. We will break down the steps involved in simplifying this expression and provide a clear explanation of the process.

Understanding the Expression

The given expression is $\frac{12 y^7}{18 y^{-3}}$. To simplify this expression, we need to understand the properties of exponents and how to manipulate them. The expression consists of two terms: $12 y^7$ and $18 y^{-3}$. The exponent $7$ indicates that the variable $y$ is raised to the power of $7$, while the exponent $-3$ indicates that the variable $y$ is raised to the power of $-3$.

Step 1: Simplify the Numerator and Denominator

To simplify the expression, we need to simplify the numerator and denominator separately. The numerator is $12 y^7$, and the denominator is $18 y^{-3}$. We can simplify the numerator by factoring out the greatest common factor (GCF) of the coefficients, which is $12$. Similarly, we can simplify the denominator by factoring out the GCF of the coefficients, which is $18$.

import sympy as sp
y = sp.symbols('y')

expr = (12y**7) / (18y**-3)

numerator = sp.simplify(12y**7)
denominator = sp.simplify(18y**-3)

Step 2: Apply the Quotient Rule for Exponents

Now that we have simplified the numerator and denominator, we can apply the quotient rule for exponents. The quotient rule states that when dividing two powers with the same base, we subtract the exponents. In this case, we have $y^7$ in the numerator and $y^{-3}$ in the denominator. We can apply the quotient rule by subtracting the exponents:

$$\frac{y^7}{y^{-3}} = y^{7-(-3)} = y^{10}$$

# Apply the quotient rule for exponents
result = sp.simplify(numerator / denominator)

Step 3: Simplify the Expression

Now that we have applied the quotient rule, we can simplify the expression further. We can simplify the expression by canceling out any common factors between the numerator and denominator. In this case, we have $y^{10}$ in the numerator and $y^{10}$ in the denominator. We can cancel out the common factor $y^{10}$:

$$\frac{12 y^7}{18 y^{-3}} = \frac{2}{3 y^{10}}$$

# Simplify the expression
final_result = sp.simplify(result)

Conclusion

In this article, we simplified the algebraic expression $\frac{12 y^7}{18 y^{-3}}$ assuming $y \neq 0$. We broke down the steps involved in simplifying the expression and provided a clear explanation of the process. We applied the quotient rule for exponents and simplified the expression further by canceling out any common factors between the numerator and denominator. The final simplified expression is $\frac{2}{3 y^{10}}$.

Answer

Introduction

In our previous article, we simplified the algebraic expression $\frac{12 y^7}{18 y^{-3}}$ assuming $y \neq 0$. We broke down the steps involved in simplifying the expression and provided a clear explanation of the process. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying algebraic expressions.

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. For example, $\frac{y^7}{y^{-3}} = y^{7-(-3)} = y^{10}$.

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

A: To apply the quotient rule for exponents, you need to follow these steps:

1. Identify the base and exponents in the numerator and denominator.
2. Subtract the exponents in the denominator from the exponents in the numerator.
3. Simplify the resulting expression.

Q: What is the greatest common factor (GCF) of the coefficients?

A: The greatest common factor (GCF) of the coefficients is the largest number that divides both coefficients without leaving a remainder. For example, the GCF of 12 and 18 is 6.

Q: How do I simplify the numerator and denominator?

A: To simplify the numerator and denominator, you need to follow these steps:

1. Factor out the greatest common factor (GCF) of the coefficients.
2. Simplify the resulting expression.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{2}{3 y^{10}}$.

Q: Can I simplify an expression with variables in the denominator?

A: Yes, you can simplify an expression with variables in the denominator. To do this, you need to follow the same steps as before, but you need to be careful when subtracting the exponents.

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

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

• Forgetting to apply the quotient rule for exponents.
• Not simplifying the numerator and denominator.
• Not canceling out common factors between the numerator and denominator.
• Not following the order of operations.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises. You can also use online resources, such as math websites and apps, to help you practice and learn.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in simplifying algebraic expressions. We covered topics such as the quotient rule for exponents, simplifying the numerator and denominator, and common mistakes to avoid. We hope that this guide has been helpful in your understanding of simplifying algebraic expressions.

Additional Resources

• Khan Academy: Algebraic Expressions
• Mathway: Algebraic Expressions
• Wolfram Alpha: Algebraic Expressions

Answer

The final answer is $\boxed{\frac{2}{3 y^{10}}}$.