Question

the sun produces $3.9 \times 10^{33}$ ergs per second of radiant energy. how much energy does it produce in one month ($2.6 \times 10^{6}$ seconds)?

Explanation:

Step1: Recall the formula for total energy

To find the total energy produced in a month, we multiply the energy produced per second by the number of seconds in a month. The formula is \( \text{Total Energy} = \text{Energy per second} \times \text{Time in seconds} \).

Step2: Substitute the given values

We know that the energy per second is \( 3.9 \times 10^{33} \) ergs and the time in seconds is \( 2.6 \times 10^{6} \) seconds. So we substitute these values into the formula:
\( (3.9 \times 10^{33}) \times (2.6 \times 10^{6}) \)

Step3: Multiply the coefficients and the powers of 10 separately

First, multiply the coefficients \( 3.9 \) and \( 2.6 \): \( 3.9 \times 2.6 = 10.14 \)
Then, multiply the powers of 10 using the rule \( a^m \times a^n = a^{m + n} \): \( 10^{33} \times 10^{6} = 10^{33 + 6} = 10^{39} \)

Step4: Combine the results

Now, combine the coefficient and the power of 10: \( 10.14 \times 10^{39} \)
But we need to express this in scientific notation. We know that \( 10.14 = 1.014 \times 10^{1} \), so:
\( 1.014 \times 10^{1} \times 10^{39} = 1.014 \times 10^{1 + 39} = 1.014 \times 10^{40} \)

Answer:

The sun produces \( 1.014 \times 10^{40} \) ergs of energy in one month.