Question
Error Analysis Your friend says that the product of 2.1 times 10 Superscript 7 and 4 times 10 Superscript negative 5 is 8.4 times 10 Superscript negative 2. What is the correct product? What mistake did your friend likely make?
Question content area bottom
Part 1
left parenthesis 2.1 times 10 Superscript 7 right parenthesis left parenthesis 4 times 10 Superscript negative 5 right parenthesisequals
enter your response here
(Use scientific notation. Use the multiplication symbol in the math palette as needed.)
Question content area bottom
Part 1
left parenthesis 2.1 times 10 Superscript 7 right parenthesis left parenthesis 4 times 10 Superscript negative 5 right parenthesisequals
enter your response here
(Use scientific notation. Use the multiplication symbol in the math palette as needed.)
Answers
GPT-4o mini
To find the product of \( (2.1 \times 10^7) \) and \( (4 \times 10^{-5}) \), we can use the property of multiplication in scientific notation, which states:
\[
(a \times 10^m)(b \times 10^n) = (a \times b) \times 10^{(m + n)}
\]
### Step 1: Multiply the coefficients
First, we multiply the coefficients \( 2.1 \) and \( 4 \):
\[
2.1 \times 4 = 8.4
\]
### Step 2: Add the exponents
Next, we add the exponents of the powers of ten:
\[
7 + (-5) = 2
\]
### Step 3: Combine results
Now, we combine the results:
\[
(2.1 \times 10^7) \times (4 \times 10^{-5}) = 8.4 \times 10^2
\]
### Final Result
The correct product is:
\[
8.4 \times 10^2
\]
### Mistake Analysis
Your friend stated that the product was \( 8.4 \times 10^{-2} \). The mistake they likely made was in the addition of the exponents. They may have incorrectly calculated \( 7 + (-5) \) as \( -2 \) instead of \( 2 \).
\[
(a \times 10^m)(b \times 10^n) = (a \times b) \times 10^{(m + n)}
\]
### Step 1: Multiply the coefficients
First, we multiply the coefficients \( 2.1 \) and \( 4 \):
\[
2.1 \times 4 = 8.4
\]
### Step 2: Add the exponents
Next, we add the exponents of the powers of ten:
\[
7 + (-5) = 2
\]
### Step 3: Combine results
Now, we combine the results:
\[
(2.1 \times 10^7) \times (4 \times 10^{-5}) = 8.4 \times 10^2
\]
### Final Result
The correct product is:
\[
8.4 \times 10^2
\]
### Mistake Analysis
Your friend stated that the product was \( 8.4 \times 10^{-2} \). The mistake they likely made was in the addition of the exponents. They may have incorrectly calculated \( 7 + (-5) \) as \( -2 \) instead of \( 2 \).