Question
What is the result of multiplying a number in scientific notation by a given value?
Answers
GPT-4o mini
To multiply a number in scientific notation by a given value, you'll follow these general steps:
1. **Express both numbers in the same format.** If the given value is not in scientific notation, you may want to convert it into scientific notation for easier multiplication.
2. **Multiply the coefficients.** If the scientific notation is represented as \( a \times 10^b \), where \( a \) is the coefficient and \( b \) is the exponent:
- If the given value is also in scientific notation \( c \times 10^d \), multiply the coefficients: \( a \times c \).
- If the given value is a standard decimal number, just treat it as a coefficient \( c \) and multiply it with \( a \).
3. **Add the exponents.** If you multiplied two scientific numbers:
- \( 10^b \) and \( 10^d \), the result will be \( 10^{(b + d)} \).
4. **Combine the results.** The final result will be formed as:
- \( (a \times c) \times 10^{(b + d)} \)
5. **Adjust if necessary.** Ensure that the coefficient is in the proper scientific notation format (between 1 and 10). If it is not, you may need to adjust the coefficient and change the exponent accordingly.
**Example:**
If you want to multiply \( 3.5 \times 10^4 \) by \( 2 \):
1. The value \( 2 \) can be expressed in scientific notation as \( 2.0 \times 10^0 \).
2. Multiply the coefficients: \( 3.5 \times 2.0 = 7.0 \).
3. Add the exponents: \( 4 + 0 = 4 \).
4. Combine: \( 7.0 \times 10^4 \).
Since \( 7.0 \) is within the range for scientific notation, the final answer is:
\[ 7.0 \times 10^4 \]
1. **Express both numbers in the same format.** If the given value is not in scientific notation, you may want to convert it into scientific notation for easier multiplication.
2. **Multiply the coefficients.** If the scientific notation is represented as \( a \times 10^b \), where \( a \) is the coefficient and \( b \) is the exponent:
- If the given value is also in scientific notation \( c \times 10^d \), multiply the coefficients: \( a \times c \).
- If the given value is a standard decimal number, just treat it as a coefficient \( c \) and multiply it with \( a \).
3. **Add the exponents.** If you multiplied two scientific numbers:
- \( 10^b \) and \( 10^d \), the result will be \( 10^{(b + d)} \).
4. **Combine the results.** The final result will be formed as:
- \( (a \times c) \times 10^{(b + d)} \)
5. **Adjust if necessary.** Ensure that the coefficient is in the proper scientific notation format (between 1 and 10). If it is not, you may need to adjust the coefficient and change the exponent accordingly.
**Example:**
If you want to multiply \( 3.5 \times 10^4 \) by \( 2 \):
1. The value \( 2 \) can be expressed in scientific notation as \( 2.0 \times 10^0 \).
2. Multiply the coefficients: \( 3.5 \times 2.0 = 7.0 \).
3. Add the exponents: \( 4 + 0 = 4 \).
4. Combine: \( 7.0 \times 10^4 \).
Since \( 7.0 \) is within the range for scientific notation, the final answer is:
\[ 7.0 \times 10^4 \]