prove (a+b)^3 = a^3 + 3a^2b+ 3ab^2 + b^3... using model.?

Question

^= raise to the power.
ex.= a^3= "a" raised to the power 3.
plz help..
how can i make a model of this..??

Answer 1

poster #1 has the right idea ...


to elaborate a bit, make 8 "blocks"
((physical ... use wood or cardboard & tape ...
styrofoam is tough to work with to get "clean" edges))

[by "side" I mean "axis" or "dimension]

one a cube with each side "a"
volume = aaa

[for now forget about notation ("x", *, ^, whatever]

one a cube with each side "b"
volume = bbb

three "blocks" with 2 sides = a & one side=b
volume = aab ........ each [total = 3aab]

three "blocks" with 2 sides = b & one side=a
volume = abb ........ each [total = 3abb]

assemble these 8 pieces and you have a cube
with each side = "a+b"

volume = (a+b)(a+b)(a+b) = (a+b)^3

you *KNOW* it will work

Answer 2

it is like (a+b)^2 * (a+b) = (a^2+2ab+b^2)*(a+b)=a^3+2a^2b+ab^2+a^b+2ab^2+b^3 = a^3+3a^2b+3ab^2+b^3

hope that is what you need, can't realise what MODEL is..

you can always find koeficijents using pascals triangle:

for ^1: 1
for ^2: 1 2 1
for ^3: 1 3 3 1
for ^4: 1 4 6 4 1
and this is how you get it: in each "column" first write 1, and the next number you get from summing up the 2 above..
for ex: i've got 1331 like: 1, and than the two numbers above (1 and 2) - their sum is 3, and then next 2 numbers including one in last (2 and 1 in this case) - their sum is also 3, and in the and write 1 always. you've got now:
1331 - and just add a and b:
1*a^3 + 3*a^2b + 3*ab^2 + b^3

Answer 3

you multiply
(a+b) (a+b) = (a+b)^2 =a^2+2ab+b^2
(a+b)^2*(a+b) = (a^2+2ab+b^2)(a+b) = a^3+3ab^2 +3a^2b+b^3

model calculate 12^3 put a=10 b=2 , use the formula

12^3 = (10+2)^3 = 1000+120+600+8=1728

You can achieve this without a paper!!

Answer 4

Have you ever thought about the "geometry" behind FOILing?

(a + b)^2 = a^2 +2ab + b^2

Think of a line segment, broken into two unequal lenghts (a and b). Create a square with these lengths

aaaaaaaaaaaaaabbbbbb
a
a
a
a
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
b

Draw the rectangle it forms, and perpendicular lines that cut the rectangle into four pieces who's areas are a^2 (upper left), b^2 (lower right) and 2 pieces whose area are each ab (upper right and lower left).

That's the 2D version. Now, add another
aaaaaaaaaaaaaaabbbbbb to make a cube and you'll find the volumes inside that cube to be the pieces that make up the expansion of (a+b)^3

Once you can visualize it, you'll never forget how to FOIL again !! (I wish I had learned that in high school ) :)

Answer 5

Use a wooden cube of side a +b made of parts: 2 smaller

cubes side a and side b , 3 sticks length a and square end

b by b, 3 sticks length b and square end a by a.

Answer 6

(a - b)(a^2 + a*b + b^2)

Answer 7

a^3 + 3a^2b+ 3ab^2 + b^3 = 2ab(a+b) + a^2(a+b) + b^2(a+b) = 2ab(a+b) + (a+b)(a^2+b^2) = (a+b)(a^2+b^2+2ab)
=(a+b)[(a+b)^2]=(a+b)^3

Answer 8

(a+b)^3 = ( a+b)(a+b)^2
=( a+b) ( a^2 + 2ab +b^2)
= a^3+2a^2b+2ab^2 +a^2b+2ab^2+b^3
= a^3 +3a^2b +23ab^2 +b^3

Answer 9

find the rule. (a+b)^n=a^n+na^(n-1)b+nab^(n-1) +...and so an and so forth.
the key here is to realize that (a+b)^3=(a+b)*(a+b)^2
and (a+b)^n= (a+b)*(a+b)^(n-1) and (a+b)^(n-1)=(a+b)*(a+b)^(n-2) and so on and so forth
i hope i made myself understood ;)

Answer 10

a cube with sides of length (a+b).