Question
Bethany wants to estimate the value of (4.296 times 10 Superscript 11) (1.8614 times 10 Superscript negative 14). Which statement about the estimate is true?
The value will be greater than 1 because 4.296 times 10 Superscript 11 is a very large number and multiplication always increases the size of a number.
The value will be greater than 1 because 4 times 2 = 8 and 8 is larger than 1.
The value will be less than 1 because (4.296 times 10 Superscript 11) (1.8614 times 10 Superscript negative 14) will be a negative number. All negative numbers are less than one.
The number will be less than 1 because when adding the exponents, 11 + (negative 14) = negative 3, and a number in scientific notation with a negative exponent is less than 1.
The value will be greater than 1 because 4.296 times 10 Superscript 11 is a very large number and multiplication always increases the size of a number.
The value will be greater than 1 because 4 times 2 = 8 and 8 is larger than 1.
The value will be less than 1 because (4.296 times 10 Superscript 11) (1.8614 times 10 Superscript negative 14) will be a negative number. All negative numbers are less than one.
The number will be less than 1 because when adding the exponents, 11 + (negative 14) = negative 3, and a number in scientific notation with a negative exponent is less than 1.
Answers
GPT-4o mini
To determine the correct statement about the estimate of \((4.296 \times 10^{11}) (1.8614 \times 10^{-14})\), let's analyze the multiplication step-by-step.
1. **Multiplicative Nature**: When multiplying numbers in scientific notation, you multiply the coefficients and add the exponents.
- Coefficient: \(4.296 \times 1.8614 \approx 8\) (for estimation, we can round to \(4 \times 2 = 8\)).
- Exponents: \(11 + (-14) = 11 - 14 = -3\).
Therefore, combining these gives:
\[
8 \times 10^{-3}
\]
2. **Interpreting the Result**:
- The expression \(8 \times 10^{-3}\) can also be written as \(0.008\), which is less than 1.
Now, let's look at the statements:
1. **"The value will be greater than 1 because 4.296 times 10 Superscript 11 is a very large number and multiplication always increases the size of a number."**
- **False**. While \(4.296 \times 10^{11}\) is very large, the multiplication with \(1.8614 \times 10^{-14}\) has a significant negative exponent, which reduces the magnitude.
2. **"The value will be greater than 1 because 4 times 2 = 8 and 8 is larger than 1."**
- **False**. Although \(8\) is greater than \(1\), the presence of the negative exponent causes the overall result to be less than \(1\).
3. **"The value will be less than 1 because (4.296 times 10 Superscript 11) (1.8614 times 10 Superscript negative 14) will be a negative number. All negative numbers are less than one."**
- **False**. The product is not negative; it is positive but less than 1.
4. **"The number will be less than 1 because when adding the exponents, 11 + (negative 14) = negative 3, and a number in scientific notation with a negative exponent is less than 1."**
- **True**. This correctly concludes that the overall product is in the form \(8 \times 10^{-3}\), which is less than 1.
Thus, the correct answer is:
**The number will be less than 1 because when adding the exponents, 11 + (negative 14) = negative 3, and a number in scientific notation with a negative exponent is less than 1.**
1. **Multiplicative Nature**: When multiplying numbers in scientific notation, you multiply the coefficients and add the exponents.
- Coefficient: \(4.296 \times 1.8614 \approx 8\) (for estimation, we can round to \(4 \times 2 = 8\)).
- Exponents: \(11 + (-14) = 11 - 14 = -3\).
Therefore, combining these gives:
\[
8 \times 10^{-3}
\]
2. **Interpreting the Result**:
- The expression \(8 \times 10^{-3}\) can also be written as \(0.008\), which is less than 1.
Now, let's look at the statements:
1. **"The value will be greater than 1 because 4.296 times 10 Superscript 11 is a very large number and multiplication always increases the size of a number."**
- **False**. While \(4.296 \times 10^{11}\) is very large, the multiplication with \(1.8614 \times 10^{-14}\) has a significant negative exponent, which reduces the magnitude.
2. **"The value will be greater than 1 because 4 times 2 = 8 and 8 is larger than 1."**
- **False**. Although \(8\) is greater than \(1\), the presence of the negative exponent causes the overall result to be less than \(1\).
3. **"The value will be less than 1 because (4.296 times 10 Superscript 11) (1.8614 times 10 Superscript negative 14) will be a negative number. All negative numbers are less than one."**
- **False**. The product is not negative; it is positive but less than 1.
4. **"The number will be less than 1 because when adding the exponents, 11 + (negative 14) = negative 3, and a number in scientific notation with a negative exponent is less than 1."**
- **True**. This correctly concludes that the overall product is in the form \(8 \times 10^{-3}\), which is less than 1.
Thus, the correct answer is:
**The number will be less than 1 because when adding the exponents, 11 + (negative 14) = negative 3, and a number in scientific notation with a negative exponent is less than 1.**