Simplify The Expression:$\[ \frac{2x^2y^4 \cdot 4x^2y^4 \cdot 3x}{3x^{-3}y^2} \\]

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 delve into the world of algebraic manipulation and explore the steps involved in simplifying a complex expression. We will use the given expression as a case study and break it down into manageable parts, applying various algebraic rules and techniques to arrive at the final simplified form.

The Given Expression

The expression we will be working with is:

$$\frac{2x^2y^4 \cdot 4x^2y^4 \cdot 3x}{3x^{-3}y^2}$$

This expression involves multiplication and division of variables with exponents, making it a perfect candidate for simplification.

Step 1: Multiply the Numerators

To simplify the expression, we start by multiplying the numerators together. This involves multiplying the coefficients (2, 4, and 3) and adding the exponents of the variables (x and y).

$$2 \cdot 4 \cdot 3 = 24$$

$$x^2 \cdot x^2 \cdot x = x^{2+2+1} = x^5$$

$$y^4 \cdot y^4 = y^{4+4} = y^8$$

So, the numerator becomes:

$$24x^5y^8$$

Step 2: Simplify the Denominator

Next, we simplify the denominator by applying the rule for dividing variables with exponents. When dividing variables with the same base, we subtract the exponents.

$$3x^{-3}y^2 = \frac{3y^2}{x^3}$$

Step 3: Divide the Numerator by the Denominator

Now that we have simplified the numerator and denominator, we can divide the numerator by the denominator. This involves dividing the coefficients and subtracting the exponents of the variables.

$$\frac{24x^5y^8}{\frac{3y^2}{x^3}} = \frac{24x^5y^8 \cdot x^3}{3y^2}$$

$$= \frac{24x^8y^8}{3y^2}$$

Step 4: Simplify the Expression Further

We can simplify the expression further by canceling out common factors in the numerator and denominator. In this case, we can cancel out a factor of y^2 from the numerator and denominator.

$$\frac{24x^8y^8}{3y^2} = \frac{24x^8y^6}{3}$$

Step 5: Final Simplification

Finally, we can simplify the expression by dividing the coefficient (24) by the denominator (3).

$$\frac{24x^8y^6}{3} = 8x^8y^6$$

And there you have it! The final simplified expression is:

$$8x^8y^6$$

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, we were able to simplify a complex expression involving multiplication and division of variables with exponents. We hope this guide has been helpful in understanding the process of algebraic manipulation and has provided a comprehensive overview of the techniques involved.

Frequently Asked Questions

• Q: What is the rule for multiplying variables with exponents? A: When multiplying variables with the same base, we add the exponents.
• Q: What is the rule for dividing variables with exponents? A: When dividing variables with the same base, we subtract the exponents.
• Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, start by simplifying the numerator and denominator separately, and then divide the numerator by the denominator.

Additional Resources

• Khan Academy: Algebraic Manipulation
• Mathway: Algebraic Expression Simplifier
• Wolfram Alpha: Algebraic Expression Simplifier

Final Thoughts

Simplifying algebraic expressions is a crucial skill for any math enthusiast. By mastering the techniques outlined in this article, you will be able to simplify complex expressions with ease and confidence. Remember to always follow the rules for multiplying and dividing variables with exponents, and don't be afraid to simplify expressions further by canceling out common factors. Happy simplifying!

Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics, and it can be a challenging task for many students. In our previous article, we provided a comprehensive guide to simplifying algebraic expressions, including step-by-step instructions and examples. However, we understand that sometimes, it's not enough to just read about a concept; you need to ask questions and get answers.

In this article, we'll address some of the most frequently asked questions about algebraic expression simplification. Whether you're a student, a teacher, or simply someone who wants to brush up on their math skills, this Q&A guide is for you.

Q: What is the rule for multiplying variables with exponents?

A: When multiplying variables with the same base, we add the exponents. For example, if we have $x^2 \cdot x^3$, the result is $x^{2+3} = x^5$.

Q: What is the rule for dividing variables with exponents?

A: When dividing variables with the same base, we subtract the exponents. For example, if we have $\frac{x^2}{x^3}$, the result is $x^{2-3} = x^{-1}$.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, start by simplifying the numerator and denominator separately, and then divide the numerator by the denominator. For example, if we have $\frac{2x^2y^4 \cdot 4x^2y^4 \cdot 3x}{3x^{-3}y^2}$, we can simplify it by following these steps:

1. Multiply the numerators together: $2 \cdot 4 \cdot 3 = 24$ and $x^2 \cdot x^2 \cdot x = x^5$ and $y^4 \cdot y^4 = y^8$.
2. Simplify the denominator: $3x^{-3}y^2 = \frac{3y^2}{x^3}$.
3. Divide the numerator by the denominator: $\frac{24x^5y^8}{\frac{3y^2}{x^3}} = \frac{24x^5y^8 \cdot x^3}{3y^2}$.
4. Simplify the expression further: $\frac{24x^5y^8 \cdot x^3}{3y^2} = \frac{24x^8y^8}{3y^2}$.
5. Cancel out common factors: $\frac{24x^8y^8}{3y^2} = \frac{24x^8y^6}{3}$.
6. Simplify the expression further: $\frac{24x^8y^6}{3} = 8x^8y^6$.

Q: What is the difference between simplifying an algebraic expression and solving an equation?

A: Simplifying an algebraic expression involves reducing the expression to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true. For example, if we have the equation $2x^2 + 3x - 1 = 0$, we need to solve for x, but if we have the expression $2x^2 + 3x - 1$, we can simplify it by combining like terms.

Q: How do I know when to simplify an algebraic expression?

A: You should simplify an algebraic expression whenever you encounter one in a problem or equation. Simplifying expressions can help you:

• Make calculations easier
• Reduce the complexity of an equation
• Identify patterns and relationships between variables
• Solve equations more efficiently

Q: Can I simplify an algebraic expression that has a variable in the denominator?

A: Yes, you can simplify an algebraic expression that has a variable in the denominator. However, you need to be careful when dividing variables with exponents. For example, if we have $\frac{x^2}{x^3}$, the result is $x^{2-3} = x^{-1}$.

Q: How do I simplify an algebraic expression that has a fraction in the numerator or denominator?

A: To simplify an algebraic expression that has a fraction in the numerator or denominator, you can follow these steps:

1. Simplify the fraction separately.
2. Multiply or divide the fraction by the rest of the expression.
3. Simplify the resulting expression.

For example, if we have $\frac{\frac{2x^2y^4}{3}}{x^3y^2}$, we can simplify it by following these steps:

1. Simplify the fraction in the numerator: $\frac{2x^2y^4}{3} = \frac{2x^2y^4}{3}$.
2. Multiply the fraction by the rest of the expression: $\frac{\frac{2x^2y^4}{3}}{x^3y^2} = \frac{2x^2y^4}{3x^3y^2}$.
3. Simplify the resulting expression: $\frac{2x^2y^4}{3x^3y^2} = \frac{2x^{-1}y^2}{3}$.

Q: Can I simplify an algebraic expression that has a negative exponent?

A: Yes, you can simplify an algebraic expression that has a negative exponent. When a variable has a negative exponent, it means that the variable is in the denominator. For example, if we have $x^{-2}$, the result is $\frac{1}{x^2}$.

Q: How do I simplify an algebraic expression that has a zero exponent?

A: When a variable has a zero exponent, it means that the variable is equal to 1. For example, if we have $x^0$, the result is 1.

Q: Can I simplify an algebraic expression that has a variable with a fractional exponent?

A: Yes, you can simplify an algebraic expression that has a variable with a fractional exponent. When a variable has a fractional exponent, it means that the variable is raised to a power that is a fraction. For example, if we have $x^{\frac{1}{2}}$, the result is $\sqrt{x}$.

Q: How do I simplify an algebraic expression that has a variable with a negative fractional exponent?

A: When a variable has a negative fractional exponent, it means that the variable is in the denominator and raised to a power that is a fraction. For example, if we have $x^{-\frac{1}{2}}$, the result is $\frac{1}{\sqrt{x}}$.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By mastering the techniques outlined in this article, you will be able to simplify complex expressions with ease and confidence. Remember to always follow the rules for multiplying and dividing variables with exponents, and don't be afraid to simplify expressions further by canceling out common factors. Happy simplifying!

Additional Resources

• Khan Academy: Algebraic Manipulation
• Mathway: Algebraic Expression Simplifier
• Wolfram Alpha: Algebraic Expression Simplifier

Final Thoughts

Simplifying algebraic expressions is a crucial skill for any math enthusiast. By mastering the techniques outlined in this article, you will be able to simplify complex expressions with ease and confidence. Remember to always follow the rules for multiplying and dividing variables with exponents, and don't be afraid to simplify expressions further by canceling out common factors. Happy simplifying!