Question

How do you use pascals triangle to expand `(d + 4)^7`?

Answer 1

Pascal's triangle can be used to give the simple coefficients of the terms `d^(7-i)4^i` for `i=0 to 7`.
`(d+4)^7 =` `d^7+28d^6+336d^5+2240d^4+8960d^3+21504d^2+28672d+16384`

Explanation:

Pascal's Triangle (first 8 rows)
#color(white)("XXX"){: (0," | ",1,,,,,,,), (1," | ",1,1,,,,,,), (2," | ",1,2,1,,,,,), (3," | ",1,3,3,1,,,,), (4," | ",1,4,6,4,1,,,), (5," | ",1,5,10,10,5,1,,), (6," | ",1,6,15,20,15,6,1,), (7," | ",1,7,21,35,35,21,7,1) :}#

`(d+4)^7`
`=1d^7 4^0 +7d^6 4^1+21d^5 4^2+35d^4 4^3+35d^3 4^4+21d^2 4^5+7d^1 4^6 + 1d^0 4^7`

...of course there is still quite a bit of multiplying within the individual terms; the following table might help"
#color(white)("XXXXXX"){: ("exponent"," | ",4^("exponent")), (0, " | ", 1), (1, " | ", 4), (2, " | ", 16), (3, " | ", 64), (4, " | ", 256), (5, " | ", 1024), (6, " | ", 4096), (7, " | ", 16384) :}#

(...I used a spreadsheet to evaluate each term for the answer shown above)