Choose The Correct Simplification Of $\left(4 X^3 Y^4 Z^5\right)\left(5 X^4 Y^5 Z^3\right$\].A. $20 X^{12} Y^{20} Z^{15}$B. $9 X^7 Y^9 Z^8$C. $9 X^{12} Y^{20} Z^{15}$D. $20 X^7 Y^9 Z^8$

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 explore the process of simplifying algebraic expressions, with a focus on the correct simplification of the expression $\left(4 x^3 y^4 z^5\right)\left(5 x^4 y^5 z^3\right)$. We will examine the different options provided and determine the correct simplification.

Understanding Exponents

Before we dive into the simplification process, it's essential to understand the concept of exponents. Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ represents $x \cdot x \cdot x$, and $y^4$ represents $y \cdot y \cdot y \cdot y$. When we multiply two numbers with the same base, we add their exponents. For example, $x^3 \cdot x^4 = x^{3+4} = x^7$.

Simplifying the Expression

Now that we have a solid understanding of exponents, let's simplify the given expression $\left(4 x^3 y^4 z^5\right)\left(5 x^4 y^5 z^3\right)$. To simplify this expression, we will multiply the coefficients (the numbers in front of the variables) and add the exponents of the variables.

Step 1: Multiply the Coefficients

The coefficients of the two expressions are 4 and 5, respectively. To multiply these coefficients, we simply multiply them together: $4 \cdot 5 = 20$.

Step 2: Add the Exponents

Now, let's add the exponents of the variables. We have $x^3$ and $x^4$, so we add their exponents: $3+4=7$. Similarly, we have $y^4$ and $y^5$, so we add their exponents: $4+5=9$. Finally, we have $z^5$ and $z^3$, so we add their exponents: $5+3=8$.

Step 3: Write the Simplified Expression

Now that we have multiplied the coefficients and added the exponents, we can write the simplified expression: $20 x^7 y^9 z^8$.

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options provided:

A. $20 x^{12} y^{20} z^{15}$

B. $9 x^7 y^9 z^8$

C. $9 x^{12} y^{20} z^{15}$

D. $20 x^7 y^9 z^8$

Based on our simplification, we can see that option D is the correct answer.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for students and professionals alike. By understanding the concept of exponents and following the steps outlined in this article, we can simplify complex expressions and arrive at the correct answer. In this case, the correct simplification of the expression $\left(4 x^3 y^4 z^5\right)\left(5 x^4 y^5 z^3\right)$ is $20 x^7 y^9 z^8$.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid:

• Incorrectly multiplying coefficients: Make sure to multiply the coefficients correctly, without forgetting any terms.
• Incorrectly adding exponents: Make sure to add the exponents correctly, without forgetting any terms.
• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

1. Simplify the expression $\left(2 x^2 y^3 z^4\right)\left(3 x^4 y^5 z^2\right)$.
2. Simplify the expression $\left(4 x^3 y^2 z^5\right)\left(5 x^4 y^3 z^3\right)$.
3. Simplify the expression $\left(2 x^2 y^4 z^3\right)\left(3 x^5 y^2 z^4\right)$.

Final Thoughts

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

A: The first step in simplifying an algebraic expression is to identify the coefficients and variables. Coefficients are the numbers in front of the variables, and variables are the letters or symbols that represent the unknown values.

Q: How do I multiply coefficients?

A: To multiply coefficients, you simply multiply the numbers together. For example, if you have the expression $2x^2y^3z^4$ and you multiply it by $3x^4y^5z^2$, you would multiply the coefficients $2$ and $3$ together to get $6$.

Q: How do I add exponents?

A: To add exponents, you add the exponents of the variables together. For example, if you have the expression $x^2y^3z^4$ and you multiply it by $x^4y^5z^2$, you would add the exponents of the variables together to get $x^{2+4}y^{3+5}z^{4+2} = x^6y^8z^6$.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an expression. The order of operations is:

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

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

A: To simplify an expression with multiple variables, you can use the same steps as before. First, identify the coefficients and variables. Then, multiply the coefficients together and add the exponents of the variables together.

Q: What is the difference between a coefficient and a variable?

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents an unknown value.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you can use the rule that $a^{-n} = \frac{1}{a^n}$. For example, if you have the expression $x^{-2}$, you can rewrite it as $\frac{1}{x^2}$.

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

A: The final step in simplifying an algebraic expression is to write the simplified expression in the simplest form possible. This means that you should have eliminated any parentheses, combined like terms, and written the expression in a way that is easy to read and understand.

Q: Why is it important to simplify algebraic expressions?

A: Simplifying algebraic expressions is important because it helps you to:

• Evaluate expressions more easily
• Solve equations more easily
• Understand the relationships between variables
• Make calculations more efficient

Q: Can I use a calculator to simplify algebraic expressions?

A: Yes, you can use a calculator to simplify algebraic expressions. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

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 multiply coefficients
• Forgetting to add exponents
• Not following the order of operations
• Not simplifying expressions fully
• Not checking work by hand

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

• Working through practice problems
• Using online resources and tools
• Asking a teacher or tutor for help
• Joining a study group or math club
• Practicing with real-world applications