To calculate the total energy generated by the Sun 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 \( E = 4 \times 10^{26} \) joules/second
- Seconds in a year \( T = 3.15 \times 10^7 \) seconds/year
The total energy generated in a year \( E_{total} \) can be calculated as:
\[ E_{total} = E \times T \]
Substituting the values:
\[ E_{total} = (4 \times 10^{26}) \times (3.15 \times 10^7) \]
Now, we can multiply the coefficients (4 and 3.15) and then add the exponents of 10:
\[ E_{total} = (4 \times 3.15) \times 10^{26 + 7} \]
Calculating the coefficient:
\[ 4 \times 3.15 = 12.6 \]
Now we add the exponents:
\[ 10^{26 + 7} = 10^{33} \]
Now, we can express the total energy:
\[ E_{total} = 12.6 \times 10^{33} \]
To express the answer in scientific notation where the coefficient is between 1 and 10, we rewrite 12.6:
\[ 12.6 = 1.26 \times 10^1 \]
Thus,
\[ E_{total} = 1.26 \times 10^1 \times 10^{33} = 1.26 \times 10^{34} \]
Therefore, the total energy generated by the Sun in a year is:
\[ \boxed{1.26 \times 10^{34}} \text{ joules} \]