Please show the work and answer!!!

Question

1. When adding two numbers, such as 123 and 423, care is taken to first line them up and then add like digits. How does expanding this expression to [(1 × 102) + (2 × 101) + (3 × 100)] + [(4 × 102) + (2 × 101) + (3 × 100)] make the operation more like a polynomial addition problem?

Answer 1

Assuming you are saying 102 = 10^2, 101 = 10^1, 100 = 10^0 (as the "^" denotes the power symbol)...

let's break this down.

We know that 123 + 143 = 266.

As you have already pointed out:

1. 123 = 1*10^2 + 2*10^1 + 3*10^0
2. 143 = 1*10^2 + 4*10^1 + 3*10^0

And so 123 + 143 = (1*10^2 + 2*10^1 + 3*10^0) + (1*10^2 + 4*10^1 + 3*10^0).

We can rearrange the equation like so:

(1*10^2 + 1*10^2) + (2*10^1 + 4*10^1) + (3*10^0 + 3*10^0)

As you can see there are multiply exponents with the same base and power. Hence, this can be simplified to

2*10^2 + 6*10^1 + 6*10^0

which is a polynomial addition problem