To calculate the total energy generated by the sun in a year, we can use the formula:
\[ \text{Total Energy} = \text{Energy per second} \times \text{Number of seconds in a year} \]
Given:
- Energy per second (E) = \(4 \times 10^{26}\) joules
- Number of seconds in a year (t) = \(3.15 \times 10^7\) seconds
Now, we can multiply these two values:
\[ \text{Total Energy} = (4 \times 10^{26}) \times (3.15 \times 10^7) \]
First, we can multiply the coefficients (4 and 3.15):
\[ 4 \times 3.15 = 12.6 \]
Next, we can multiply the powers of ten:
\[ 10^{26} \times 10^{7} = 10^{(26 + 7)} = 10^{33} \]
Now, combine the results:
\[ \text{Total Energy} = 12.6 \times 10^{33} \]
To express this in scientific notation, we need to convert 12.6 to a number between 1 and 10. We can do this by rewriting 12.6 as:
\[ 12.6 = 1.26 \times 10^1 \]
Now substituting this back, we get:
\[ \text{Total Energy} = (1.26 \times 10^{1}) \times 10^{33} = 1.26 \times 10^{34} \]
Thus, the total energy generated by the sun in a year is:
\[ \boxed{1.26 \times 10^{34}} \text{ joules} \]