Lets see again :

11575
125
—————–on addition
117625

139
507
2197
——————-on adding
1442897

the term *3 digit number* is too broad.
Lets break it down to *managable* types of 3 digit numbers.

A) The simple cases : eg cube of (105)>>> this can be handled by a process of “mental” calculations .
here the base number is 100 :
In 105 , the excess over the base 100 is 5
Take this 5 and do 3 x 5 =15
The first digit of the answer **FROM THE LEFT** is 1
so that 1 , and attach 15 to it ; you get 115

Next multiply the 15 obtained earlier by the 8excess* 5 ; 15 x 5 = 75
Attach this to 115
So you have 11575
Now take the cube of the excess 5. you get 125 ; attach this also;

11575
125
——————- now addd these
1157625 >>>>>>>is the cube of 105
{ * check the correctness of the answer by the method of dropping
the NINES }
Happy ?

Lets take the cube of (113) :
First digit of the answer from the LEFT is 1
Excess x 3 = 39 ; So 139
Next 39x the excess = 39×13
=(40-1)x 13
= 520 -13
= 507
The cube of 13 = 2197

Place these in this form :
1 3 9
5 0 7
2 1 9 7
——————-Add
Answer >>>>>>>>>>>>>>>> 1 4 4 2 8 9 7

WE MAY CONTINUE, AFTER THERE IS A FEEDBACK FROM YOU
With best wishes.

Here let a=4 ; b=6 and c =2

Find the following
ac + b 2 = 8 + 36 = 44 ——(z)
c 3 = 2 3 = 8 ————————————–> 8
3bc 2 = 3X6X4 =72 ——————————> 2 CF =7
3c X (z) = 6 X 44 = 264 + 7=271 —————-> 1 CF =27
b(6ac+b 2 ) = 6(48+36) =504 +27=531 ———-> 1 CF = 53
3a X (z) = 12 X 44 = 528 + 53 =581 ————-> 1 CF =58
3a 2 b = 3 X 16 X 6 = 288 +58= 346 ————-> 6 CF =34
a 3 =64 +34 = 98 ———————————> 8
—————————————————-> 9

so 462 3 = 98611128

Use formula (a+b)³= a³ + 3a²b + 3ab² + b³

(100+15)³=100³+ 3×100²x15+3x100x15² +15³

=1000000+750000+67500+3375
ans= 1820875

(125*7)/5= 175

3) 5 : 7
175 : ?

?=245

4) 5 : 7
245 : ?

? = 343

only do it upto 4 step

now
step1 step2 step3 step4
125 175 245 343
350 490 (Double of step 2 and step 3)
—————————————–

add & carry forward Right To Left

answer 185193