Simplify: $\[ (-2x)\left(2x^6\right)^2 \\]\[$\square\$\]Suggested Tutorials: - Learn It: Simplify Expressions Using The Product

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Understanding the Problem

Simplifying expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this problem, we're given the expression ${ (-2x)\left(2x6\right)2 }${\square\$}$, and we need to simplify it using the product rule.

The Product Rule

The product rule states that when we multiply two or more expressions, we can simplify the result by multiplying the coefficients and adding the exponents of the variables. In this case, we have two expressions: −2x-2x and (2x6)2\left(2x^6\right)^2. We'll start by simplifying the second expression using the power rule.

Simplifying the Second Expression

The power rule states that when we raise a power to another power, we can multiply the exponents. In this case, we have (2x6)2\left(2x^6\right)^2. Using the power rule, we can simplify this expression as follows:

(2x6)2=22â‹…x6â‹…2=4x12\left(2x^6\right)^2 = 2^2 \cdot x^{6 \cdot 2} = 4x^{12}

Multiplying the Expressions

Now that we've simplified the second expression, we can multiply it with the first expression, −2x-2x. Using the product rule, we can multiply the coefficients and add the exponents of the variables:

−2x⋅4x12=−8x1+12=−8x13-2x \cdot 4x^{12} = -8x^{1+12} = -8x^{13}

Final Answer

Therefore, the simplified expression is −8x13\boxed{-8x^{13}}.

Tips and Tricks

  • When simplifying expressions, always start by simplifying the expressions inside the parentheses.
  • Use the power rule to simplify expressions with exponents.
  • Use the product rule to multiply expressions and simplify the result.
  • Always check your work by plugging in simple values for the variables.

Suggested Tutorials

  • Learn It: Simplify expressions using the product rule
  • Practice It: Simplify expressions with exponents and coefficients
  • Watch It: Video tutorials on simplifying expressions

Discussion Category

  • Mathematics
  • Algebra
  • Simplifying expressions
  • Product rule
  • Power rule

Related Topics

  • Simplifying expressions with fractions
  • Simplifying expressions with decimals
  • Simplifying expressions with negative exponents
  • Simplifying expressions with variables in the denominator

Conclusion

Simplifying expressions is an essential skill in mathematics, and it's crucial to understand the rules and techniques involved. By following the product rule and using the power rule, we can simplify complex expressions and arrive at the final answer. Remember to always check your work and practice simplifying expressions to become proficient in this skill.

Frequently Asked Questions

Q: What is the product rule in mathematics?

A: The product rule is a fundamental concept in mathematics that states that when we multiply two or more expressions, we can simplify the result by multiplying the coefficients and adding the exponents of the variables.

Q: How do I simplify expressions using the product rule?

A: To simplify expressions using the product rule, start by simplifying the expressions inside the parentheses. Then, use the power rule to simplify expressions with exponents. Finally, use the product rule to multiply the expressions and simplify the result.

Q: What is the power rule in mathematics?

A: The power rule is a fundamental concept in mathematics that states that when we raise a power to another power, we can multiply the exponents.

Q: How do I simplify expressions with exponents using the power rule?

A: To simplify expressions with exponents using the power rule, multiply the exponents of the variables. For example, if we have (2x6)2\left(2x^6\right)^2, we can simplify it as follows:

(2x6)2=22â‹…x6â‹…2=4x12\left(2x^6\right)^2 = 2^2 \cdot x^{6 \cdot 2} = 4x^{12}

Q: What is the difference between the product rule and the power rule?

A: The product rule and the power rule are two fundamental concepts in mathematics that are used to simplify expressions. The product rule is used to multiply expressions and simplify the result, while the power rule is used to simplify expressions with exponents.

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, plug in simple values for the variables and evaluate the expression. This will help you ensure that your simplification is correct.

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

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

  • Forgetting to simplify expressions inside the parentheses
  • Not using the power rule to simplify expressions with exponents
  • Not multiplying the coefficients and adding the exponents of the variables when using the product rule
  • Not checking your work when simplifying expressions

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through practice problems and exercises. You can also use online resources and video tutorials to help you learn and practice simplifying expressions.

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

A: Simplifying expressions has many real-world applications, including:

  • Calculating the area and perimeter of shapes
  • Determining the cost of goods and services
  • Solving problems in physics and engineering
  • Analyzing data and making predictions

Conclusion

Simplifying expressions is an essential skill in mathematics, and it's crucial to understand the rules and techniques involved. By following the product rule and using the power rule, we can simplify complex expressions and arrive at the final answer. Remember to always check your work and practice simplifying expressions to become proficient in this skill.

Suggested Tutorials

  • Learn It: Simplify expressions using the product rule
  • Practice It: Simplify expressions with exponents and coefficients
  • Watch It: Video tutorials on simplifying expressions

Discussion Category

  • Mathematics
  • Algebra
  • Simplifying expressions
  • Product rule
  • Power rule

Related Topics

  • Simplifying expressions with fractions
  • Simplifying expressions with decimals
  • Simplifying expressions with negative exponents
  • Simplifying expressions with variables in the denominator

Additional Resources

  • Online resources and video tutorials on simplifying expressions
  • Practice problems and exercises on simplifying expressions
  • Real-world applications of simplifying expressions