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.
The sun generates \( 4 \times 10^{26} \) joules of energy every second, and there are \( 3.15 \times 10^7 \) seconds in a year.
To calculate the total energy generated in a year:
\[ \text{Total Energy} = \text{Energy per second} \times \text{Seconds in a year} \]
Substituting the values:
\[ \text{Total Energy} = (4 \times 10^{26}) \times (3.15 \times 10^{7}) \]
First, we can multiply the coefficients (4 and 3.15) and then handle the powers of ten:
\[ 4 \times 3.15 = 12.6 \]
Next, we add the exponents from the powers of ten:
\[ 10^{26} \times 10^{7} = 10^{(26 + 7)} = 10^{33} \]
Now, we have:
\[ \text{Total Energy} = 12.6 \times 10^{33} \]
To express this in proper scientific notation, we need the coefficient to be between 1 and 10. We can rewrite \( 12.6 \) as \( 1.26 \times 10^1 \):
\[ 12.6 \times 10^{33} = 1.26 \times 10^{34} \]
Finally, the energy the sun generates in a year, expressed in scientific notation, is:
\[ \boxed{1.26 \times 10^{34}} \text{ joules} \]
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