Simplify The Expression: $\frac{6 X^{-4}}{4 X^2 Y^{-5}}$

Feb 28, 2025 by ADMIN 57 views

$Simplify the Expression: $\frac{6 x^{-4}}{4 x^2 y^{-5}}$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying the given expression: $\frac{6 x^{-4}}{4 x^2 y^{-5}}$ . We will use the properties of exponents and follow the order of operations to simplify the expression.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics 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 $\frac{6 x^{-4}}{4 x^2 y^{-5}}$ , we need to follow the order of operations (PEMDAS):

Parentheses: There are no parentheses in the expression, so we move on to the next step.
Exponents: We can simplify the exponents by using the properties of exponents. Specifically, we can use the rule that states $a^{-n} = \frac{1}{a^n}$ .
Multiplication and Division: We can simplify the expression by dividing the numerator and denominator by their greatest common factor (GCF).
Addition and Subtraction: There are no addition or subtraction operations in the expression, so we can move on to the final step.

Applying the Properties of Exponents

Using the properties of exponents, we can rewrite the expression as:

\frac{6 x^{-4}}{4 x^2 y^{-5}} = \frac{6}{4} \cdot \frac{1}{x^4} \cdot \frac{1}{x^2} \cdot \frac{1}{y^5}

Simplifying the Numerator and Denominator

We can simplify the numerator and denominator by dividing them by their greatest common factor (GCF). In this case, the GCF of 6 and 4 is 2.

\frac{6}{4} = \frac{3}{2}

Simplifying the Expression

Now we can simplify the expression by combining the fractions:

\frac{3}{2} \cdot \frac{1}{x^4} \cdot \frac{1}{x^2} \cdot \frac{1}{y^5} = \frac{3}{2x^6y^5}

Conclusion

In this article, we simplified the expression $\frac{6 x^{-4}}{4 x^2 y^{-5}}$ using the properties of exponents and the order of operations. We applied the rule that states $a^{-n} = \frac{1}{a^n}$ and simplified the numerator and denominator by dividing them by their greatest common factor (GCF). The final simplified expression is $\frac{3}{2x^6y^5}$ .

Tips and Tricks

When simplifying expressions, always follow the order of operations (PEMDAS).
Use the properties of exponents to simplify expressions.
Identify the greatest common factor (GCF) of the numerator and denominator and divide them by it.
Combine fractions by multiplying the numerators and denominators.

Common Mistakes to Avoid

Not following the order of operations (PEMDAS).
Not using the properties of exponents to simplify expressions.
Not identifying the greatest common factor (GCF) of the numerator and denominator.
Not combining fractions by multiplying the numerators and denominators.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, simplifying expressions is used to solve problems involving motion, energy, and momentum. In engineering, simplifying expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.

Final Thoughts

Simplifying algebraic expressions is a fundamental skill in mathematics, and it requires a deep understanding of the properties of exponents and the order of operations. By following the steps outlined in this article, you can simplify even the most complex expressions and apply the skills to real-world problems.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

Khan Academy: Algebra
MIT OpenCourseWare: Algebra
Wolfram Alpha: Algebra Calculator

Introduction

In our previous article, we simplified the expression $\frac{6 x^{-4}}{4 x^2 y^{-5}}$ using the properties of exponents and the order of operations. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

Q1: What is the order of operations (PEMDAS)?

A1: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q2: How do I simplify expressions with negative exponents?

A2: To simplify expressions with negative exponents, we can use the rule that states $a^{-n} = \frac{1}{a^n}$ . For example, in the expression $x^{-4}$ , we can rewrite it as $\frac{1}{x^4}$ .

Q3: What is the greatest common factor (GCF)?

A3: The greatest common factor (GCF) of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCF of 6 and 4 is 2.

Q4: How do I simplify expressions with multiple variables?

A4: To simplify expressions with multiple variables, we can use the properties of exponents and the order of operations. For example, in the expression $\frac{6 x^{-4} y^{-5}}{4 x^2 y^{-3}}$ , we can simplify it by using the properties of exponents and the order of operations.

Q5: What are some common mistakes to avoid when simplifying expressions?

A5: Some common mistakes to avoid when simplifying expressions include:

Not following the order of operations (PEMDAS).
Not using the properties of exponents to simplify expressions.
Not identifying the greatest common factor (GCF) of the numerator and denominator.
Not combining fractions by multiplying the numerators and denominators.

Q6: How do I apply simplifying expressions to real-world problems?

A6: Simplifying expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, simplifying expressions is used to solve problems involving motion, energy, and momentum. In engineering, simplifying expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.

Tips and Tricks

Always follow the order of operations (PEMDAS).
Use the properties of exponents to simplify expressions.
Identify the greatest common factor (GCF) of the numerator and denominator.
Combine fractions by multiplying the numerators and denominators.

Common Mistakes to Avoid

Not following the order of operations (PEMDAS).
Not using the properties of exponents to simplify expressions.
Not identifying the greatest common factor (GCF) of the numerator and denominator.
Not combining fractions by multiplying the numerators and denominators.