Fully Simplify: ${-3 X^3\left(16 X^4 Y^2\right)}$ Answer Attempt 1 Out Of 2:${ \square }$

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Understanding the Problem

When simplifying algebraic expressions, it's essential to understand the rules of exponents and how to apply them to various mathematical operations. In this article, we'll focus on simplifying the given expression: ${-3 x^3\left(16 x^4 y^2\right)}$. Our goal is to simplify this expression using the rules of exponents and basic algebraic operations.

The Rules of Exponents

Before we dive into simplifying the given expression, let's review the rules of exponents. The rules of exponents state that when multiplying two or more variables with the same base, we add their exponents. On the other hand, when dividing two or more variables with the same base, we subtract their exponents.

Simplifying the Expression

Now that we've reviewed the rules of exponents, let's simplify the given expression. To simplify the expression ${-3 x^3\left(16 x^4 y^2\right)}$, we need to apply the rules of exponents and basic algebraic operations.

When multiplying two or more variables with the same base, we add their exponents. In this case, we have x3x^3 and x4x^4, which have the same base (xx). Therefore, we can add their exponents to get x3+4=x7x^{3+4} = x^7.

Similarly, when multiplying two or more variables with the same base, we add their exponents. In this case, we have y2y^2, which has the same base (yy). Therefore, we can add its exponent to get y2+0=y2y^{2+0} = y^2.

Now that we've simplified the variables, let's simplify the coefficients. The coefficient of the expression is −3-3, which is a constant. Therefore, we can multiply it by the simplified variables to get −3x7y2-3x^7y^2.

The Final Answer

In conclusion, the simplified expression is −3x7y2\boxed{-3x^7y^2}.

Discussion and Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it requires a deep understanding of the rules of exponents and basic algebraic operations. In this article, we've simplified the given expression using the rules of exponents and basic algebraic operations. We've also reviewed the rules of exponents and provided a step-by-step guide on how to simplify the expression.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid. Some of these mistakes include:

  • Not applying the rules of exponents correctly: When multiplying two or more variables with the same base, we add their exponents. However, when dividing two or more variables with the same base, we subtract their exponents.
  • Not simplifying the coefficients correctly: The coefficients of an expression are the numerical values that are multiplied by the variables. When simplifying the coefficients, we need to multiply them by the simplified variables.
  • Not checking the final answer: Before providing the final answer, we need to check it to ensure that it's correct.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. Some of these applications include:

  • Physics and Engineering: Algebraic expressions are used to describe the laws of physics and engineering. Simplifying these expressions is essential to understand the underlying principles.
  • Computer Science: Algebraic expressions are used in computer science to describe algorithms and data structures. Simplifying these expressions is essential to optimize the performance of computer programs.
  • Economics: Algebraic expressions are used in economics to describe economic models and theories. Simplifying these expressions is essential to understand the underlying principles.

Conclusion

Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions about simplifying algebraic expressions.

Q: What are the rules of exponents?

A: The rules of exponents state that when multiplying two or more variables with the same base, we add their exponents. On the other hand, when dividing two or more variables with the same base, we subtract their exponents.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to apply the rules of exponents and basic algebraic operations. Here are the steps to follow:

  1. Identify the variables and coefficients: Identify the variables and coefficients in the expression.
  2. Apply the rules of exponents: Apply the rules of exponents to simplify the variables.
  3. Simplify the coefficients: Simplify the coefficients by multiplying them by the simplified variables.
  4. Check the final answer: Check the final answer to ensure that it's correct.

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

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

  • Not applying the rules of exponents correctly: When multiplying two or more variables with the same base, we add their exponents. However, when dividing two or more variables with the same base, we subtract their exponents.
  • Not simplifying the coefficients correctly: The coefficients of an expression are the numerical values that are multiplied by the variables. When simplifying the coefficients, we need to multiply them by the simplified variables.
  • Not checking the final answer: Before providing the final answer, we need to check it to ensure that it's correct.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has numerous real-world applications. Some of these applications include:

  • Physics and Engineering: Algebraic expressions are used to describe the laws of physics and engineering. Simplifying these expressions is essential to understand the underlying principles.
  • Computer Science: Algebraic expressions are used in computer science to describe algorithms and data structures. Simplifying these expressions is essential to optimize the performance of computer programs.
  • Economics: Algebraic expressions are used in economics to describe economic models and theories. Simplifying these expressions is essential to understand the underlying principles.

Q: How can I practice simplifying algebraic expressions?

A: There are several ways to practice simplifying algebraic expressions. Some of these include:

  • Solving algebraic expression problems: Practice solving algebraic expression problems to improve your skills.
  • Using online resources: Use online resources such as algebraic expression calculators and worksheets to practice simplifying algebraic expressions.
  • Working with a tutor or teacher: Work with a tutor or teacher to practice simplifying algebraic expressions and get feedback on your work.

Q: What are some tips for simplifying algebraic expressions?

A: Some tips for simplifying algebraic expressions include:

  • Read the problem carefully: Read the problem carefully to understand what's being asked.
  • Apply the rules of exponents correctly: Apply the rules of exponents correctly to simplify the variables.
  • Simplify the coefficients correctly: Simplify the coefficients by multiplying them by the simplified variables.
  • Check the final answer: Check the final answer to ensure that it's correct.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill in mathematics, and it requires a deep understanding of the rules of exponents and basic algebraic operations. By following the rules of exponents and basic algebraic operations, we can simplify complex algebraic expressions and provide accurate solutions to mathematical problems.