Start by displacing the numbers into the given formula:
{{F=}\frac{Gm_1m_2}{r^2}}= {\frac {(7 \times10^{-11)} \times (6 \times 10^{24}) \times (7 \times 10^{22)}}{ (4 \times 10^8 )^2}}
Numerator calculations:
Multiply all the non-exponent integers in the numerator: {7\times 6 \times 7}= 294.
Using exponent rules, we know that numbers with the same bases multiplied means the exponents must be added together:
{x^ax^bx^c= x^{a+b+c}}
So, {10^{-11}10^{24}10^{22}}= {10^{-11+24+22}}= {10^{35}}
Which sets our numerator value at {294\times 10^{35}}
However, we know that the rules of standard form dictate:
{A \times 10^n}
where { 1\le A < 10} and n is an integer.
Therefore, we must adjust our value to become: {2.94 \times 10^{37}}
Denominator Calculations:
Once again we use exponent rules, particularly “Power to a Power” to say:
{(a^m)^n} = {a^{m\times n}}
Hence, calculating-
{(4^1)^2}= {4^{1\times 2}} = {4^2}= 16
{(10^8)^2} = {10^{8\times 2}}= {10^{16}}
Giving the answer {16\times 10^{16}}
And remembering to adjust this into appropriate Standard Form measures, it becomes {1.6\times 10^{17}}
Putting the Numerator and Denominator of our fraction together it becomes:
{F= \frac {2.94 \times 10^{37}}{1.6 \times 10^17}}
If you focus on the question, you’ll see they’ve given you the help of calculating the first part of our fraction, and we now know that {\frac{2.94}{1.6}}= 1.8365
Our final calculation then is simple, with the knowledge of the Quotient Rule:
{\frac{a^m}{a^n}}= {a^{m-n}}
Making {\frac{10^{37}}{10^{17}}}= {10^{37-17}}= {10^{20}}
Our final answer is: {1.8365\times 10^{20}} , which is B.