Simplify The Expression:${ \frac{36 X^{12} Y^{-5}}{2 X^6 Y} }$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the given expression: $\frac{36 x^{12} y^{-5}}{2 x^6 y}$. We will break down the process into manageable steps, making it easy to understand and follow along.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what we're dealing with. The given expression is a fraction, where the numerator is $36 x^{12} y^{-5}$ and the denominator is $2 x^6 y$. Our goal is to simplify this expression by combining like terms and reducing the fraction to its simplest form.

Step 1: Simplify the Numerator

The numerator of the expression is $36 x^{12} y^{-5}$. We can start by simplifying this term. To do this, we need to factor out the greatest common factor (GCF) of the coefficients and the variables.

36 x^{12} y^{-5} = 36 \cdot x^{12} \cdot y^{-5}

The GCF of the coefficients is 36, and the GCF of the variables is $x^{12} y^{-5}$. We can factor out these terms to simplify the numerator.

36 x^{12} y^{-5} = 36 \cdot (x^{12} \cdot y^{-5})

Step 2: Simplify the Denominator

The denominator of the expression is $2 x^6 y$. We can simplify this term by factoring out the GCF of the coefficients and the variables.

2 x^6 y = 2 \cdot x^6 \cdot y

The GCF of the coefficients is 2, and the GCF of the variables is $x^6 y$. We can factor out these terms to simplify the denominator.

2 x^6 y = 2 \cdot (x^6 \cdot y)

Step 3: Simplify the Fraction

Now that we have simplified the numerator and the denominator, we can simplify the fraction by dividing the numerator by the denominator.

\frac{36 x^{12} y^{-5}}{2 x^6 y} = \frac{36 \cdot (x^{12} \cdot y^{-5})}{2 \cdot (x^6 \cdot y)}

To simplify the fraction, we can cancel out common factors between the numerator and the denominator.

\frac{36 \cdot (x^{12} \cdot y^{-5})}{2 \cdot (x^6 \cdot y)} = \frac{36}{2} \cdot \frac{x^{12}}{x^6} \cdot \frac{y^{-5}}{y}

Step 4: Simplify the Coefficients

The coefficients in the numerator and the denominator are 36 and 2, respectively. We can simplify these coefficients by dividing them.

\frac{36}{2} = 18

Step 5: Simplify the Variables

The variables in the numerator and the denominator are $x^{12}$ and $x^6$, respectively, and $y^{-5}$ and $y$, respectively. We can simplify these variables by combining like terms.

\frac{x^{12}}{x^6} = x^{12-6} = x^6

\frac{y^{-5}}{y} = y^{-5-1} = y^{-6}

Step 6: Simplify the Expression

Now that we have simplified the coefficients and the variables, we can simplify the expression by combining the simplified terms.

\frac{36 x^{12} y^{-5}}{2 x^6 y} = 18 \cdot x^6 \cdot y^{-6}

Conclusion

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Introduction

In our previous article, we explored the process of simplifying algebraic expressions. We took a step-by-step approach to simplify the expression: $\frac{36 x^{12} y^{-5}}{2 x^6 y}$. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

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

A: The first step in simplifying an algebraic expression is to identify the greatest common factor (GCF) of the coefficients and the variables.

Q: How do I simplify the numerator of an algebraic expression?

A: To simplify the numerator of an algebraic expression, you need to factor out the GCF of the coefficients and the variables.

Q: How do I simplify the denominator of an algebraic expression?

A: To simplify the denominator of an algebraic expression, you need to factor out the GCF of the coefficients and the variables.

Q: What is the difference between simplifying an algebraic expression and reducing a fraction?

A: Simplifying an algebraic expression involves combining like terms and reducing the fraction to its simplest form. Reducing a fraction involves dividing the numerator and the denominator by their greatest common divisor (GCD).

Q: Can I simplify an algebraic expression with negative exponents?

A: Yes, you can simplify an algebraic expression with negative exponents. To do this, you need to rewrite the expression with positive exponents and then simplify.

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

A: To simplify an algebraic expression with multiple variables, you need to identify the GCF of the coefficients and the variables, and then factor out the GCF.

Q: Can I simplify an algebraic expression with fractions?

A: Yes, you can simplify an algebraic expression with fractions. To do this, you need to simplify the fractions separately and then combine them.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to check if the expression can be simplified further.

Example Questions

Example 1: Simplify the expression $\frac{24 x^3 y^2}{6 x^2 y}$.

To simplify this expression, we need to identify the GCF of the coefficients and the variables. The GCF of the coefficients is 6, and the GCF of the variables is $x^2 y$. We can factor out these terms to simplify the expression.

\frac{24 x^3 y^2}{6 x^2 y} = \frac{24}{6} \cdot \frac{x^3}{x^2} \cdot \frac{y^2}{y}

Simplifying the coefficients and the variables, we get:

\frac{24}{6} = 4
\frac{x^3}{x^2} = x
\frac{y^2}{y} = y

Combining the simplified terms, we get:

\frac{24 x^3 y^2}{6 x^2 y} = 4x y

Example 2: Simplify the expression $\frac{48 x^4 y^{-3}}{8 x^2 y^{-1}}$.

To simplify this expression, we need to identify the GCF of the coefficients and the variables. The GCF of the coefficients is 8, and the GCF of the variables is $x^2 y^{-1}$. We can factor out these terms to simplify the expression.

\frac{48 x^4 y^{-3}}{8 x^2 y^{-1}} = \frac{48}{8} \cdot \frac{x^4}{x^2} \cdot \frac{y^{-3}}{y^{-1}}

Simplifying the coefficients and the variables, we get:

\frac{48}{8} = 6
\frac{x^4}{x^2} = x^2
\frac{y^{-3}}{y^{-1}} = y^{-2}

Combining the simplified terms, we get:

\frac{48 x^4 y^{-3}}{8 x^2 y^{-1}} = 6x^2 y^{-2}

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, we can simplify complex expressions and arrive at their simplest form. We hope this Q&A guide has been helpful in answering your questions and providing a better understanding of simplifying algebraic expressions.