What is the value of $142,857 \times 7^2$?

Do you recognize that $1/7=0.\overline{142857}?$ If so, you will recognize that $142,857 \cdot 7 = 999,999,$ so $142,857 \cdot 7^2=(1,000,000-1)\cdot 7=6,999,993$

Multipling by $7^2=49$ is the same as multiplying by $50$ then subtracting one copy.

So $$142,857*7^2=142,857*50-142857$$. Now, multipling by 50 is the same as multiplying by 100 then dividing by 2. so we have $$(142,857*100)/2-142857$$. Multiplying by 100 is just adding 2 0s: $$14285700/2-142857$$ Dividing by 2 is easy to do by hand: $$7142850-142857$$ And finally, subtracting two numbers is easy to do by hand: $$6999993$$ Hopefully that's actually right, because I'm not checking :)

If you recognize that those are the decimal digits of $1/7 = .142857142857142857...$, then you realize that $1000000/7$ must be 142857+0.142857, so 142857 must be (100,000 - 1)/7, and you're on your way.