What Does $3^3 \div 3^{-1}$ Equal?A. 9B. 1 9 \frac{1}{9} 9 1 ​ C. 1 81 \frac{1}{81} 81 1 ​ D. 81

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Understanding Exponents and Division

In mathematics, exponents are a shorthand way of representing repeated multiplication. When we see an expression like $3^3$, it means 3 multiplied by itself 3 times, or $3 \times 3 \times 3$. This can be simplified to 27. On the other hand, $3^{-1}$ represents the reciprocal of 3, or 1 divided by 3, which is $\frac{1}{3}$.

The Division of Exponents

When we divide two numbers with the same base, we subtract the exponents. In this case, we have $3^3 \div 3^{-1}$. To evaluate this expression, we need to apply the rule for dividing exponents with the same base.

Applying the Rule for Dividing Exponents

The rule for dividing exponents with the same base is:

am÷an=am−na^m \div a^n = a^{m-n}

where $a$ is the base, and $m$ and $n$ are the exponents.

In our case, we have:

33÷3−1=33−(−1)3^3 \div 3^{-1} = 3^{3-(-1)}

Simplifying the Exponent

Now, let's simplify the exponent:

3−(−1)=3+1=43-(-1) = 3+1 = 4

So, we have:

33÷3−1=343^3 \div 3^{-1} = 3^4

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it:

34=3×3×3×3=813^4 = 3 \times 3 \times 3 \times 3 = 81

Therefore, the final answer is:

The Final Answer

33÷3−1=813^3 \div 3^{-1} = 81

Conclusion

In this article, we have seen how to evaluate the expression $3^3 \div 3^{-1}$. We applied the rule for dividing exponents with the same base and simplified the exponent to get the final answer. This is an important concept in mathematics, and it can be applied to a wide range of problems.

Common Mistakes

When evaluating expressions with exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not applying the rule for dividing exponents: Make sure to apply the rule for dividing exponents with the same base.
  • Not simplifying the exponent: Make sure to simplify the exponent before evaluating the expression.
  • Not evaluating the expression correctly: Make sure to evaluate the expression correctly after simplifying the exponent.

Practice Problems

Here are some practice problems to help you reinforce your understanding of exponents and division:

  • 24÷2−22^4 \div 2^{-2}

  • 53÷5−15^3 \div 5^{-1}

  • 42÷4−14^2 \div 4^{-1}

Answer Key

Here are the answers to the practice problems:

  • 24÷2−2=24−(−2)=26=642^4 \div 2^{-2} = 2^{4-(-2)} = 2^6 = 64

  • 53÷5−1=53−(−1)=54=6255^3 \div 5^{-1} = 5^{3-(-1)} = 5^4 = 625

  • 42÷4−1=42−(−1)=43=644^2 \div 4^{-1} = 4^{2-(-1)} = 4^3 = 64

Final Thoughts

Frequently Asked Questions

In this article, we will answer some frequently asked questions about exponents and division. Whether you are a student or a teacher, these questions and answers will help you understand the concepts of exponents and division.

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

A: The rule for dividing exponents with the same base is:

am÷an=am−na^m \div a^n = a^{m-n}

where $a$ is the base, and $m$ and $n$ are the exponents.

Q: How do I simplify the exponent when dividing exponents with the same base?

A: To simplify the exponent, you need to subtract the exponents. For example, if you have:

33÷3−13^3 \div 3^{-1}

You would simplify the exponent by subtracting the exponents:

3−(−1)=3+1=43-(-1) = 3+1 = 4

So, the simplified expression is:

343^4

Q: What is the difference between $a^m \div a^n$ and $a^{m-n}$?

A: $a^m \div a^n$ is the expression for dividing two numbers with the same base, while $a^{m-n}$ is the simplified expression.

For example, if you have:

33÷3−13^3 \div 3^{-1}

The expression is $3^3 \div 3^{-1}$, but the simplified expression is $3^4$.

Q: Can I use the rule for dividing exponents with the same base for negative exponents?

A: Yes, you can use the rule for dividing exponents with the same base for negative exponents. For example, if you have:

3−3÷3−13^{-3} \div 3^{-1}

You would simplify the exponent by subtracting the exponents:

−3−(−1)=−3+1=−2-3-(-1) = -3+1 = -2

So, the simplified expression is:

3−23^{-2}

Q: How do I evaluate an expression with exponents and division?

A: To evaluate an expression with exponents and division, you need to follow these steps:

  1. Simplify the exponent by subtracting the exponents.
  2. Evaluate the simplified expression.

For example, if you have:

33÷3−13^3 \div 3^{-1}

You would simplify the exponent by subtracting the exponents:

3−(−1)=3+1=43-(-1) = 3+1 = 4

So, the simplified expression is:

343^4

You would then evaluate the simplified expression:

34=3×3×3×3=813^4 = 3 \times 3 \times 3 \times 3 = 81

Q: What are some common mistakes to avoid when evaluating expressions with exponents and division?

A: Here are some common mistakes to avoid:

  • Not applying the rule for dividing exponents: Make sure to apply the rule for dividing exponents with the same base.
  • Not simplifying the exponent: Make sure to simplify the exponent before evaluating the expression.
  • Not evaluating the expression correctly: Make sure to evaluate the expression correctly after simplifying the exponent.

Q: How can I practice evaluating expressions with exponents and division?

A: Here are some practice problems to help you reinforce your understanding of exponents and division:

  • 24÷2−22^4 \div 2^{-2}

  • 53÷5−15^3 \div 5^{-1}

  • 42÷4−14^2 \div 4^{-1}

Answer Key

Here are the answers to the practice problems:

  • 24÷2−2=24−(−2)=26=642^4 \div 2^{-2} = 2^{4-(-2)} = 2^6 = 64

  • 53÷5−1=53−(−1)=54=6255^3 \div 5^{-1} = 5^{3-(-1)} = 5^4 = 625

  • 42÷4−1=42−(−1)=43=644^2 \div 4^{-1} = 4^{2-(-1)} = 4^3 = 64

Final Thoughts

In conclusion, evaluating expressions with exponents and division requires a good understanding of the rules for exponents and division. By applying these rules and simplifying the exponent, we can evaluate expressions like $3^3 \div 3^{-1}$ and get the final answer. With practice and patience, you can become proficient in evaluating expressions with exponents and division.