Question
The sun generates about 4×10 to the power of 26
joules of energy every second. There are about 3.15×10 to the power of 7
seconds in a year. About how many joules of energy does the sun generate in a year? Write the answer in scientific notation. Express the answer to the exact decimal place.(1 point)
joules of energy every second. There are about 3.15×10 to the power of 7
seconds in a year. About how many joules of energy does the sun generate in a year? Write the answer in scientific notation. Express the answer to the exact decimal place.(1 point)
Answers
GPT-4o mini
To find out how many joules of energy the sun generates in a year, we can multiply the energy generated per second by the number of seconds in a year.
Given:
- Energy generated per second by the sun = \( 4 \times 10^{26} \) joules
- Seconds in a year = \( 3.15 \times 10^{7} \) seconds
Now, we perform the multiplication:
\[
\text{Total energy in a year} = (4 \times 10^{26}) \times (3.15 \times 10^{7})
\]
We multiply the coefficients and add the exponents of 10:
1. Multiply the coefficients:
\[
4 \times 3.15 = 12.6
\]
2. Add the exponents:
\[
10^{26} \times 10^{7} = 10^{(26 + 7)} = 10^{33}
\]
Now, combine the two results:
\[
\text{Total energy} = 12.6 \times 10^{33}
\]
To express this in proper scientific notation, we need to adjust \( 12.6 \) to be between 1 and 10:
\[
12.6 = 1.26 \times 10^{1}
\]
Thus,
\[
12.6 \times 10^{33} = 1.26 \times 10^{1} \times 10^{33} = 1.26 \times 10^{34}
\]
Therefore, the total energy generated by the sun in a year is:
\[
\boxed{1.26 \times 10^{34}}
\] joules.
Given:
- Energy generated per second by the sun = \( 4 \times 10^{26} \) joules
- Seconds in a year = \( 3.15 \times 10^{7} \) seconds
Now, we perform the multiplication:
\[
\text{Total energy in a year} = (4 \times 10^{26}) \times (3.15 \times 10^{7})
\]
We multiply the coefficients and add the exponents of 10:
1. Multiply the coefficients:
\[
4 \times 3.15 = 12.6
\]
2. Add the exponents:
\[
10^{26} \times 10^{7} = 10^{(26 + 7)} = 10^{33}
\]
Now, combine the two results:
\[
\text{Total energy} = 12.6 \times 10^{33}
\]
To express this in proper scientific notation, we need to adjust \( 12.6 \) to be between 1 and 10:
\[
12.6 = 1.26 \times 10^{1}
\]
Thus,
\[
12.6 \times 10^{33} = 1.26 \times 10^{1} \times 10^{33} = 1.26 \times 10^{34}
\]
Therefore, the total energy generated by the sun in a year is:
\[
\boxed{1.26 \times 10^{34}}
\] joules.
In a given year, about 1.52×10 to the power of 10
pieces of first-class mail are delivered in the U.S. If each piece of mail carries a stamp costing $0.55, what is the total cost in dollars to deliver all the pieces? Write the answer in scientific notation, expressed to the exact decimal place.(1 point)
pieces of first-class mail are delivered in the U.S. If each piece of mail carries a stamp costing $0.55, what is the total cost in dollars to deliver all the pieces? Write the answer in scientific notation, expressed to the exact decimal place.(1 point)
GPT-4o mini
To calculate the total cost of delivering all the pieces of first-class mail, we multiply the number of pieces of mail by the cost of each stamp.
Given:
- Pieces of mail = \( 1.52 \times 10^{10} \)
- Cost of each stamp = \( 0.55 \) dollars
Now, we perform the multiplication:
\[
\text{Total cost} = (1.52 \times 10^{10}) \times (0.55)
\]
First, we multiply the coefficients:
\[
1.52 \times 0.55 = 0.836
\]
Next, we include the exponent from the scientific notation:
\[
\text{Total cost} = 0.836 \times 10^{10}
\]
To express this in proper scientific notation, we need to adjust \( 0.836 \) so that the coefficient is between 1 and 10:
\[
0.836 = 8.36 \times 10^{-1}
\]
Now, we can re-write the total cost, combining the terms:
\[
0.836 \times 10^{10} = 8.36 \times 10^{-1} \times 10^{10} = 8.36 \times 10^{(10 - 1)} = 8.36 \times 10^{9}
\]
Thus, the total cost to deliver all the pieces of first-class mail is:
\[
\boxed{8.36 \times 10^{9}} \text{ dollars.}
\]
Given:
- Pieces of mail = \( 1.52 \times 10^{10} \)
- Cost of each stamp = \( 0.55 \) dollars
Now, we perform the multiplication:
\[
\text{Total cost} = (1.52 \times 10^{10}) \times (0.55)
\]
First, we multiply the coefficients:
\[
1.52 \times 0.55 = 0.836
\]
Next, we include the exponent from the scientific notation:
\[
\text{Total cost} = 0.836 \times 10^{10}
\]
To express this in proper scientific notation, we need to adjust \( 0.836 \) so that the coefficient is between 1 and 10:
\[
0.836 = 8.36 \times 10^{-1}
\]
Now, we can re-write the total cost, combining the terms:
\[
0.836 \times 10^{10} = 8.36 \times 10^{-1} \times 10^{10} = 8.36 \times 10^{(10 - 1)} = 8.36 \times 10^{9}
\]
Thus, the total cost to deliver all the pieces of first-class mail is:
\[
\boxed{8.36 \times 10^{9}} \text{ dollars.}
\]
Question
What is 5.5×10 to the power of 4 multiplied by 2.2×10 to the power of−8 ? Write the answer in scientific notation.(1 point)
GPT-4o mini
To multiply the numbers \( 5.5 \times 10^{4} \) and \( 2.2 \times 10^{-8} \), we will multiply the coefficients and then add the exponents.
1. **Multiply the coefficients:**
\[
5.5 \times 2.2 = 12.1
\]
2. **Add the exponents:**
\[
10^{4} \times 10^{-8} = 10^{(4 + (-8))} = 10^{-4}
\]
Now, combining the results gives us:
\[
12.1 \times 10^{-4}
\]
To express this in proper scientific notation, we need the coefficient to be between 1 and 10. We can convert \( 12.1 \) as follows:
\[
12.1 = 1.21 \times 10^{1}
\]
Then we rewrite the total:
\[
12.1 \times 10^{-4} = 1.21 \times 10^{1} \times 10^{-4} = 1.21 \times 10^{(1 - 4)} = 1.21 \times 10^{-3}
\]
Thus, the final answer in scientific notation is:
\[
\boxed{1.21 \times 10^{-3}}
\]
1. **Multiply the coefficients:**
\[
5.5 \times 2.2 = 12.1
\]
2. **Add the exponents:**
\[
10^{4} \times 10^{-8} = 10^{(4 + (-8))} = 10^{-4}
\]
Now, combining the results gives us:
\[
12.1 \times 10^{-4}
\]
To express this in proper scientific notation, we need the coefficient to be between 1 and 10. We can convert \( 12.1 \) as follows:
\[
12.1 = 1.21 \times 10^{1}
\]
Then we rewrite the total:
\[
12.1 \times 10^{-4} = 1.21 \times 10^{1} \times 10^{-4} = 1.21 \times 10^{(1 - 4)} = 1.21 \times 10^{-3}
\]
Thus, the final answer in scientific notation is:
\[
\boxed{1.21 \times 10^{-3}}
\]
When multiplying two numbers in scientific notation, what operations do you use on the coefficients and the exponents of the powers of 10?(1 point)
Responses
Add the coefficients and multiply the exponents of the powers of 10.
Add the coefficients and multiply the exponents of the powers of 10.
Multiply the coefficients and multiply the exponents of the powers of 10.
Multiply the coefficients and multiply the exponents of the powers of 10.
Add the coefficients and add the exponents of the powers of 10.
Add the coefficients and add the exponents of the powers of 10.
Multiply the coefficients and add the exponents of the powers of 10.
Responses
Add the coefficients and multiply the exponents of the powers of 10.
Add the coefficients and multiply the exponents of the powers of 10.
Multiply the coefficients and multiply the exponents of the powers of 10.
Multiply the coefficients and multiply the exponents of the powers of 10.
Add the coefficients and add the exponents of the powers of 10.
Add the coefficients and add the exponents of the powers of 10.
Multiply the coefficients and add the exponents of the powers of 10.
GPT-4o mini
When multiplying two numbers in scientific notation, you **multiply the coefficients** and **add the exponents** of the powers of 10.
So the correct response is:
**Multiply the coefficients and add the exponents of the powers of 10.**
So the correct response is:
**Multiply the coefficients and add the exponents of the powers of 10.**
Multiply 0.75 by 2.8×10 to the power of−5
.(1 point)
.(1 point)
GPT-4o mini
To multiply \( 0.75 \) by \( 2.8 \times 10^{-5} \), we will first multiply the coefficient \( 0.75 \) by \( 2.8 \) and then express the result in scientific notation.
1. **Multiply the coefficients:**
\[
0.75 \times 2.8 = 2.1
\]
2. **Include the scientific notation part:**
Since there is no additional power of 10 associated with \( 0.75 \), we simply keep the power of 10 from \( 2.8 \times 10^{-5} \):
\[
2.1 \times 10^{-5}
\]
Therefore, the result of multiplying \( 0.75 \) by \( 2.8 \times 10^{-5} \) is:
\[
\boxed{2.1 \times 10^{-5}}
\]
1. **Multiply the coefficients:**
\[
0.75 \times 2.8 = 2.1
\]
2. **Include the scientific notation part:**
Since there is no additional power of 10 associated with \( 0.75 \), we simply keep the power of 10 from \( 2.8 \times 10^{-5} \):
\[
2.1 \times 10^{-5}
\]
Therefore, the result of multiplying \( 0.75 \) by \( 2.8 \times 10^{-5} \) is:
\[
\boxed{2.1 \times 10^{-5}}
\]