Pablo Simplified The Expression $\left(2^{-4}\right)^{-2}$ As Shown:$\left(2^{-4}\right)^{-2} = 2^{-4 \cdot (-2)} = 2^{-8}$Which Statement Explains Pablo's Error?A. He Should Have Added The Exponents Instead Of Multiplying Them.B. He

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Introduction

Exponents are a fundamental concept in mathematics, and understanding the rules governing their behavior is crucial for solving complex problems. In this article, we will delve into the world of exponents and analyze a common mistake made by students, including Pablo, who incorrectly simplified the expression (2−4)−2\left(2^{-4}\right)^{-2}.

The Correct Approach

To simplify the expression (2−4)−2\left(2^{-4}\right)^{-2}, we need to apply the power rule of exponents, which states that for any non-zero number aa and integers mm and nn, (am)n=am⋅n(a^m)^n = a^{m \cdot n}. Using this rule, we can rewrite the expression as follows:

(2−4)−2=2−4⋅(−2)\left(2^{-4}\right)^{-2} = 2^{-4 \cdot (-2)}

However, Pablo incorrectly simplified the expression by multiplying the exponents instead of adding them. This mistake is a common one, and it's essential to understand why it's incorrect.

The Error: Multiplying Exponents Instead of Adding

Pablo's mistake can be attributed to a fundamental misunderstanding of the exponent rules. When dealing with negative exponents, it's essential to remember that they represent reciprocals. In this case, 2−42^{-4} is equivalent to 124\frac{1}{2^4}, and (−2)(-2) is a negative exponent that represents a reciprocal.

To simplify the expression correctly, we need to apply the rule for a power raised to a power, which states that (am)n=am⋅n(a^m)^n = a^{m \cdot n}. However, when dealing with negative exponents, we need to add the exponents instead of multiplying them.

The Correct Simplification

Using the correct approach, we can simplify the expression as follows:

(2−4)−2=2−4⋅(−2)=28\left(2^{-4}\right)^{-2} = 2^{-4 \cdot (-2)} = 2^{8}

However, this is not the correct simplification. Since we are dealing with a negative exponent, we need to add the exponents instead of multiplying them. Therefore, the correct simplification is:

(2−4)−2=2−4+(−2)=2−6\left(2^{-4}\right)^{-2} = 2^{-4 + (-2)} = 2^{-6}

Conclusion

In conclusion, Pablo's error was due to a misunderstanding of the exponent rules. He incorrectly multiplied the exponents instead of adding them, which led to an incorrect simplification of the expression. By understanding the correct approach and applying the rules for negative exponents, we can simplify the expression correctly and arrive at the correct solution.

Common Mistakes and Their Corrections

To avoid making the same mistake as Pablo, it's essential to understand the common pitfalls and their corrections. Here are some common mistakes and their corrections:

  • Mistake: Multiplying exponents instead of adding them.
  • Correction: When dealing with negative exponents, add the exponents instead of multiplying them.
  • Mistake: Forgetting to apply the rule for a power raised to a power.
  • Correction: Always apply the rule for a power raised to a power, and remember to add the exponents when dealing with negative exponents.

Real-World Applications

Understanding exponent rules is crucial in various real-world applications, including:

  • Science: Exponents are used to represent large numbers and to simplify complex calculations in scientific applications.
  • Engineering: Exponents are used to represent large numbers and to simplify complex calculations in engineering applications.
  • Finance: Exponents are used to calculate interest rates and to simplify complex financial calculations.

Conclusion

Introduction

Exponents are a fundamental concept in mathematics, and understanding the rules governing their behavior is crucial for solving complex problems. In this article, we will address some common questions and answers related to exponent rules, providing a comprehensive guide to help you master this essential concept.

Q: What is the power rule of exponents?

A: The power rule of exponents states that for any non-zero number aa and integers mm and nn, (am)n=am⋅n(a^m)^n = a^{m \cdot n}. This rule allows us to simplify expressions by multiplying the exponents.

Q: How do I simplify expressions with negative exponents?

A: When dealing with negative exponents, you need to add the exponents instead of multiplying them. For example, (2−4)−2=2−4+(−2)=2−6(2^{-4})^{-2} = 2^{-4 + (-2)} = 2^{-6}.

Q: What is the rule for a power raised to a power?

A: The rule for a power raised to a power states that (am)n=am⋅n(a^m)^n = a^{m \cdot n}. This rule allows us to simplify expressions by multiplying the exponents.

Q: How do I handle exponents with the same base but different exponents?

A: When dealing with exponents with the same base but different exponents, you need to add the exponents. For example, 23⋅24=23+4=272^3 \cdot 2^4 = 2^{3 + 4} = 2^7.

Q: What is the rule for multiplying exponents with the same base?

A: The rule for multiplying exponents with the same base states that am⋅an=am+na^m \cdot a^n = a^{m + n}. This rule allows us to simplify expressions by adding the exponents.

Q: How do I handle exponents with different bases?

A: When dealing with exponents with different bases, you need to multiply the bases and add the exponents. For example, 23⋅34=(2⋅3)3+4=672^3 \cdot 3^4 = (2 \cdot 3)^{3 + 4} = 6^7.

Q: What is the rule for dividing exponents with the same base?

A: The rule for dividing exponents with the same base states that aman=am−n\frac{a^m}{a^n} = a^{m - n}. This rule allows us to simplify expressions by subtracting the exponents.

Q: How do I handle exponents with negative bases?

A: When dealing with exponents with negative bases, you need to follow the same rules as for positive bases. For example, (−2)3=−8(-2)^3 = -8.

Q: What is the rule for exponents with fractional bases?

A: The rule for exponents with fractional bases states that (am)n=am⋅n(a^m)^n = a^{m \cdot n}. This rule allows us to simplify expressions by multiplying the exponents.

Conclusion

In conclusion, understanding exponent rules is crucial for solving complex problems in mathematics and real-world applications. By addressing common questions and answers, we can provide a comprehensive guide to help you master this essential concept. Remember to apply the correct approach and avoid common mistakes to simplify expressions correctly and arrive at the correct solution.

Common Mistakes and Their Corrections

To avoid making the same mistakes as others, it's essential to understand the common pitfalls and their corrections. Here are some common mistakes and their corrections:

  • Mistake: Multiplying exponents instead of adding them.
  • Correction: When dealing with negative exponents, add the exponents instead of multiplying them.
  • Mistake: Forgetting to apply the rule for a power raised to a power.
  • Correction: Always apply the rule for a power raised to a power, and remember to add the exponents when dealing with negative exponents.

Real-World Applications

Understanding exponent rules is crucial in various real-world applications, including:

  • Science: Exponents are used to represent large numbers and to simplify complex calculations in scientific applications.
  • Engineering: Exponents are used to represent large numbers and to simplify complex calculations in engineering applications.
  • Finance: Exponents are used to calculate interest rates and to simplify complex financial calculations.

Conclusion

In conclusion, understanding exponent rules is essential for solving complex problems in mathematics and real-world applications. By applying the correct approach and avoiding common mistakes, we can simplify expressions correctly and arrive at the correct solution.