Simplify The Expression:${ \frac{-16 X^2 Y^7}{12 X^5 Y^3 Z^4} }$
===========================================================
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: . We will break down the process into manageable steps, making it easy to understand and follow along.
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 and the denominator being . Our goal is to simplify this expression by reducing it to its simplest form.
Step 1: Factor Out Common Terms
The first step in simplifying the expression is to factor out common terms from both the numerator and the denominator. In the numerator, we have , and in the denominator, we have . We can start by factoring out the common terms from the numerator and the denominator.
Numerator: -16 x^2 y^7 = -8 \* 2 \* x^2 \* y^3 \* y^4
Denominator: 12 x^5 y^3 z^4 = 12 \* x^3 \* x^2 \* y^3 \* z^4
Step 2: Cancel Out Common Factors
Now that we have factored out the common terms, we can cancel out the common factors between the numerator and the denominator. In this case, we have in the numerator and in the denominator, which can be canceled out.
Numerator: -8 \* 2 \* x^2 \* y^3 \* y^4
Denominator: 12 \* x^3 \* x^2 \* y^3 \* z^4
Canceling out common factors:
-8 * 2 = -16 (canceled out)
x^2 = x^2 (no cancellation)
y^3 = y^3 (no cancellation)
y^4 = y^4 (no cancellation)
x^3 = x^3 (no cancellation)
x^2 = x^2 (no cancellation)
z^4 = z^4 (no cancellation)
Simplified expression:
-2 * x^2 * y^4 / (x^3 * y^3 * z^4)
Step 3: Simplify the Expression Further
Now that we have canceled out the common factors, we can simplify the expression further by reducing it to its simplest form. In this case, we can simplify the expression by dividing the numerator and the denominator by their greatest common factor (GCF).
Simplified expression:
-2 \* x^2 \* y^4 / (x^3 \* y^3 \* z^4)
GCF of numerator and denominator:
x^2
Dividing numerator and denominator by GCF:
-2 * y^4 / (x * y^3 * z^4)
Step 4: Final Simplification
The final step in simplifying the expression is to simplify it further by reducing it to its simplest form. In this case, we can simplify the expression by dividing the numerator and the denominator by their greatest common factor (GCF).
Simplified expression:
-2 \* y^4 / (x \* y^3 \* z^4)
GCF of numerator and denominator:
y^3
Dividing numerator and denominator by GCF:
-2 * y / (x * z^4)
Conclusion
In this article, we have simplified the given expression: . We have broken down the process into manageable steps, making it easy to understand and follow along. By factoring out common terms, canceling out common factors, and simplifying the expression further, we have arrived at the final simplified expression: .
Frequently Asked Questions
Q: What is the final simplified expression?
A: The final simplified expression is .
Q: How do I simplify an expression?
A: To simplify an expression, you can follow these steps: factor out common terms, cancel out common factors, and simplify the expression further by reducing it to its simplest form.
Q: What is the greatest common factor (GCF)?
A: The greatest common factor (GCF) is the largest factor that divides two or more numbers without leaving a remainder.
Q: How do I find the GCF of two numbers?
A: To find the GCF of two numbers, you can list the factors of each number and find the largest factor that they have in common.
References
Further Reading
- Algebraic Manipulation: A Step-by-Step Guide
- Simplifying Expressions: A Guide for Students
- Greatest Common Factor: A Guide for Students
===========================================================
Q&A: Simplifying Expressions
Q: What is the purpose of simplifying expressions?
A: The purpose of simplifying expressions is to reduce them to their simplest form, making it easier to work with and understand the underlying mathematical concepts.
Q: How do I know when to simplify an expression?
A: You should simplify an expression when it is necessary to make the expression easier to work with, understand, or solve. This can be when the expression is complex, has multiple variables, or has fractions.
Q: What are some common mistakes to avoid when simplifying expressions?
A: Some common mistakes to avoid when simplifying expressions include:
- Not canceling out common factors
- Not simplifying fractions
- Not reducing expressions to their simplest form
- Not checking for errors in calculations
Q: How do I simplify expressions with fractions?
A: To simplify expressions with fractions, you can follow these steps:
- Factor out common terms from the numerator and denominator.
- Cancel out common factors between the numerator and denominator.
- Simplify the fraction by reducing it to its simplest form.
Q: How do I simplify expressions with variables?
A: To simplify expressions with variables, you can follow these steps:
- Identify the variables in the expression.
- Simplify the expression by combining like terms.
- Reduce the expression to its simplest form.
Q: What is the greatest common factor (GCF)?
A: The greatest common factor (GCF) is the largest factor that divides two or more numbers without leaving a remainder.
Q: How do I find the GCF of two numbers?
A: To find the GCF of two numbers, you can list the factors of each number and find the largest factor that they have in common.
Q: What is the least common multiple (LCM)?
A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common.
Q: How do I find the LCM of two numbers?
A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that they have in common.
Simplifying Expressions: Common Mistakes to Avoid
1. Not Canceling Out Common Factors
When simplifying expressions, it is essential to cancel out common factors between the numerator and denominator. Failure to do so can lead to incorrect results.
2. Not Simplifying Fractions
When simplifying expressions, it is essential to simplify fractions by reducing them to their simplest form. Failure to do so can lead to incorrect results.
3. Not Reducing Expressions to Their Simplest Form
When simplifying expressions, it is essential to reduce them to their simplest form. Failure to do so can lead to incorrect results.
4. Not Checking for Errors in Calculations
When simplifying expressions, it is essential to check for errors in calculations. Failure to do so can lead to incorrect results.
Simplifying Expressions: Tips and Tricks
1. Use a Calculator to Check Your Work
When simplifying expressions, it is a good idea to use a calculator to check your work. This can help you catch errors and ensure that your results are accurate.
2. Break Down Complex Expressions into Simpler Ones
When simplifying expressions, it is a good idea to break down complex expressions into simpler ones. This can make it easier to work with and understand the underlying mathematical concepts.
3. Use Algebraic Manipulation Techniques
When simplifying expressions, it is a good idea to use algebraic manipulation techniques such as factoring, canceling, and simplifying fractions. These techniques can help you simplify expressions and make them easier to work with.
Conclusion
In this article, we have discussed the importance of simplifying expressions and provided tips and tricks for doing so. We have also covered common mistakes to avoid and provided a step-by-step guide to simplifying expressions. By following these tips and tricks, you can simplify expressions and make them easier to work with and understand.
Frequently Asked Questions
Q: What is the final simplified expression?
A: The final simplified expression is .
Q: How do I simplify an expression?
A: To simplify an expression, you can follow these steps: factor out common terms, cancel out common factors, and simplify the expression further by reducing it to its simplest form.
Q: What is the greatest common factor (GCF)?
A: The greatest common factor (GCF) is the largest factor that divides two or more numbers without leaving a remainder.
Q: How do I find the GCF of two numbers?
A: To find the GCF of two numbers, you can list the factors of each number and find the largest factor that they have in common.