Aubrey Says That The Product Of $10^4$ And $10^{-2}$ Is $ 10 − 8 10^{-8} 1 0 − 8 [/tex]. Is She Correct? If Not, Explain Why.A. Yes, She Is Correct.B. No, The Exponent Should Be Positive 8.C. No, She Multiplied The Exponents

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When it comes to multiplying numbers with exponents, it's essential to understand the rules and how they apply to different scenarios. In this article, we'll delve into the world of exponents and explore whether Aubrey's statement about the product of $10^4$ and $10^{-2}$ is correct.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $10^4$ means 10 multiplied by itself 4 times, which is equal to 10,000. Similarly, $10^{-2}$ means 10 multiplied by itself -2 times, which is equal to 0.01.

Multiplying Numbers with Exponents

When multiplying numbers with exponents, we need to follow a specific rule. If we have two numbers with exponents, say $a^m$ and $a^n$, where $a$ is a non-zero number and $m$ and $n$ are integers, then the product of these two numbers is $a^{m+n}$. This means that we add the exponents together to get the new exponent.

Applying the Rule to Aubrey's Statement

Now that we understand the rule for multiplying numbers with exponents, let's apply it to Aubrey's statement. She claims that the product of $10^4$ and $10^{-2}$ is $10^{-8}$. To verify this, we need to multiply the two numbers using the rule we discussed earlier.

104×10−2=104+(−2)=10210^4 \times 10^{-2} = 10^{4+(-2)} = 10^2

As we can see, the product of $10^4$ and $10^{-2}$ is actually $10^2$, not $10^{-8}$. This means that Aubrey's statement is incorrect.

Why Did Aubrey Make This Mistake?

There are several reasons why Aubrey might have made this mistake. One possible reason is that she misunderstood the rule for multiplying numbers with exponents. Another reason is that she might have gotten confused with the concept of negative exponents.

Negative Exponents

Negative exponents can be a bit tricky to understand, but they're actually quite simple. A negative exponent means that we're taking the reciprocal of the number. For example, $10^{-2}$ means 1 divided by $10^2$, which is equal to 0.01.

Conclusion

In conclusion, Aubrey's statement about the product of $10^4$ and $10^{-2}$ is incorrect. The correct product is $10^2$, not $10^{-8}$. We hope that this article has helped you understand the rule for multiplying numbers with exponents and how to apply it to different scenarios.

Common Mistakes to Avoid

When working with exponents, it's essential to avoid making common mistakes. Here are a few mistakes to watch out for:

  • Not following the rule for multiplying numbers with exponents: Make sure to add the exponents together when multiplying numbers with exponents.
  • Getting confused with negative exponents: Remember that a negative exponent means taking the reciprocal of the number.
  • Not simplifying expressions: Make sure to simplify expressions by combining like terms and applying the rules for exponents.

Practice Problems

To help you practice working with exponents, we've included a few practice problems below:

  1. What is the product of $10^3$ and $10^2$?
  2. What is the product of $10^{-1}$ and $10^{-3}$?
  3. What is the product of $10^4$ and $10^{-4}$?

Answers

  1. 103+2=10510^{3+2} = 10^5

  2. 10−1−3=10−410^{-1-3} = 10^{-4}

  3. 104+(−4)=100=110^{4+(-4)} = 10^0 = 1

In this article, we'll answer some frequently asked questions about exponents. Whether you're a student, a teacher, or just someone who wants to learn more about exponents, this article is for you.

Q: What is the rule for multiplying numbers with exponents?

A: The rule for multiplying numbers with exponents is to add the exponents together. For example, if we have $a^m$ and $a^n$, then the product of these two numbers is $a^{m+n}$.

Q: What is the rule for dividing numbers with exponents?

A: The rule for dividing numbers with exponents is to subtract the exponents. For example, if we have $a^m$ and $a^n$, then the quotient of these two numbers is $a^{m-n}$.

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

A: The rule for raising a power to a power is to multiply the exponents. For example, if we have $(am)n$, then the result is $a^{m \cdot n}$.

Q: What is the rule for multiplying a number by itself?

A: The rule for multiplying a number by itself is to add 1 to the exponent. For example, if we have $a^m$, then the result of multiplying it by itself is $a^{m+1}$.

Q: What is the rule for raising a number to a negative power?

A: The rule for raising a number to a negative power is to take the reciprocal of the number. For example, if we have $a^{-m}$, then the result is $\frac{1}{a^m}$.

Q: What is the rule for raising a number to a fractional power?

A: The rule for raising a number to a fractional power is to take the nth root of the number, where n is the denominator of the fraction. For example, if we have $a^{\frac{1}{n}}$, then the result is $\sqrt[n]{a}$.

Q: What is the rule for multiplying numbers with different bases?

A: The rule for multiplying numbers with different bases is to multiply the numbers as if they had the same base, and then simplify the result. For example, if we have $2^3 \cdot 3^2$, then the result is $(2 \cdot 3)^3 = 6^3$.

Q: What is the rule for dividing numbers with different bases?

A: The rule for dividing numbers with different bases is to divide the numbers as if they had the same base, and then simplify the result. For example, if we have $2^3 \div 3^2$, then the result is $\frac{23}{32} = \frac{8}{9}$.

Q: What is the rule for raising a number to a power with a variable exponent?

A: The rule for raising a number to a power with a variable exponent is to use the power rule, which states that $(am)n = a^{m \cdot n}$. For example, if we have $(2x)y$, then the result is $2^{x \cdot y}$.

Q: What is the rule for multiplying numbers with exponents and variables?

A: The rule for multiplying numbers with exponents and variables is to multiply the numbers as if they had the same base, and then simplify the result. For example, if we have $2^x \cdot 3^y$, then the result is $(2 \cdot 3)^{x+y} = 6^{x+y}$.

Q: What is the rule for dividing numbers with exponents and variables?

A: The rule for dividing numbers with exponents and variables is to divide the numbers as if they had the same base, and then simplify the result. For example, if we have $2^x \div 3^y$, then the result is $\frac{2x}{3y}$.

We hope that this article has helped you understand the rules for working with exponents. With practice and patience, you'll become a pro at working with exponents in no time!