Select The Best Answer For The Question.Simplify $2 A^2 B^3\left(4 A^2+3 A B^2-a B\right$\].A. $8 A^4 B^5+3 A^3 B^5-2 A^3 B^4$B. $8 A^4 B^5+3 A^3 B^5+2 A^3 B^4$C. $8 A^4 B^3+6 A^3 B^5+2 A^3 B^4$D. $8 A^4 B^3+6 A^3

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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 focus on simplifying a specific algebraic expression, 2a2b3(4a2+3ab2−ab)2 a^2 b^3\left(4 a^2+3 a b^2-a b\right), and explore the different methods and techniques involved in simplifying complex expressions.

Understanding the Expression

The given expression is 2a2b3(4a2+3ab2−ab)2 a^2 b^3\left(4 a^2+3 a b^2-a b\right). To simplify this expression, we need to understand the rules of exponents and how to distribute terms inside parentheses.

Distributing Terms Inside Parentheses

When we have an expression inside parentheses, we can distribute the terms outside the parentheses to each term inside. In this case, we have the term 2a2b32 a^2 b^3 outside the parentheses, and the expression 4a2+3ab2−ab4 a^2+3 a b^2-a b inside the parentheses.

To distribute the terms, we multiply each term inside the parentheses by the term outside the parentheses. This gives us:

2a2b3⋅4a2+2a2b3⋅3ab2−2a2b3⋅ab2 a^2 b^3 \cdot 4 a^2 + 2 a^2 b^3 \cdot 3 a b^2 - 2 a^2 b^3 \cdot a b

Simplifying Each Term

Now that we have distributed the terms, we can simplify each term separately.

  • 2a2b3⋅4a2=8a4b32 a^2 b^3 \cdot 4 a^2 = 8 a^4 b^3
  • 2a2b3⋅3ab2=6a3b52 a^2 b^3 \cdot 3 a b^2 = 6 a^3 b^5
  • 2a2b3⋅ab=2a3b42 a^2 b^3 \cdot a b = 2 a^3 b^4

Combining Like Terms

Now that we have simplified each term, we can combine like terms to get the final simplified expression.

  • 8a4b3+6a3b5+2a3b48 a^4 b^3 + 6 a^3 b^5 + 2 a^3 b^4

Conclusion

In conclusion, simplifying algebraic expressions requires a clear understanding of the rules of exponents and how to distribute terms inside parentheses. By following the steps outlined in this article, we can simplify complex expressions and arrive at the final answer.

Answer

The final simplified expression is 8a4b3+6a3b5+2a3b48 a^4 b^3 + 6 a^3 b^5 + 2 a^3 b^4. This matches option C, which is the correct answer.

Discussion

This problem requires a clear understanding of the rules of exponents and how to distribute terms inside parentheses. It also requires the ability to simplify complex expressions and combine like terms.

Tips and Tricks

  • When simplifying algebraic expressions, it's essential to follow the order of operations (PEMDAS).
  • When distributing terms inside parentheses, make sure to multiply each term inside the parentheses by the term outside the parentheses.
  • When combining like terms, make sure to add or subtract the coefficients of the like terms.

Common Mistakes

  • Failing to distribute terms inside parentheses correctly.
  • Failing to simplify each term correctly.
  • Failing to combine like terms correctly.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including:

  • Science and Engineering: Simplifying algebraic expressions is essential in science and engineering, where complex equations need to be solved to understand and predict natural phenomena.
  • Computer Programming: Simplifying algebraic expressions is also essential in computer programming, where complex algorithms need to be simplified to improve efficiency and performance.
  • Finance: Simplifying algebraic expressions is also essential in finance, where complex financial models need to be simplified to make informed investment decisions.

Conclusion

Introduction

In our previous article, we explored the concept of simplifying algebraic expressions and provided a step-by-step guide on how to simplify a specific expression. In this article, we will provide a Q&A guide to help you better understand the concept and address any questions or concerns you may have.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It is a way to represent a mathematical relationship between variables and constants.

Q: What are the rules of exponents?

A: The rules of exponents are a set of rules that govern how to simplify expressions with exponents. The main rules are:

  • Product of Powers: When multiplying two powers with the same base, add the exponents.
  • Power of a Power: When raising a power to a power, multiply the exponents.
  • Power of a Product: When raising a product to a power, raise each factor to that power.

Q: How do I distribute terms inside parentheses?

A: To distribute terms inside parentheses, multiply each term inside the parentheses by the term outside the parentheses. This is done by multiplying each term inside the parentheses by the coefficient of the term outside the parentheses.

Q: What is the order of operations?

A: The order of operations is a set of rules that govern how to evaluate mathematical expressions. The order of operations is:

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

Q: How do I simplify complex expressions?

A: To simplify complex expressions, follow these steps:

  1. Distribute terms inside parentheses: Multiply each term inside the parentheses by the term outside the parentheses.
  2. Combine like terms: Add or subtract the coefficients of like terms.
  3. Simplify each term: Simplify each term by applying the rules of exponents and combining like terms.

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

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Failing to distribute terms inside parentheses correctly: Make sure to multiply each term inside the parentheses by the term outside the parentheses.
  • Failing to simplify each term correctly: Make sure to apply the rules of exponents and combine like terms.
  • Failing to combine like terms correctly: Make sure to add or subtract the coefficients of like terms.

Q: How do I apply simplifying algebraic expressions in real-world scenarios?

A: Simplifying algebraic expressions has numerous real-world applications, including:

  • Science and Engineering: Simplifying algebraic expressions is essential in science and engineering, where complex equations need to be solved to understand and predict natural phenomena.
  • Computer Programming: Simplifying algebraic expressions is also essential in computer programming, where complex algorithms need to be simplified to improve efficiency and performance.
  • Finance: Simplifying algebraic expressions is also essential in finance, where complex financial models need to be simplified to make informed investment decisions.

Conclusion

In conclusion, simplifying algebraic expressions is a critical skill that requires a clear understanding of the rules of exponents and how to distribute terms inside parentheses. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex expressions and apply them in real-world scenarios.