Fill in the squares 16x^4/36 - _ +36x^2/16= 81y^6 + 180y^3*x^5+_=

To complete the given squares, we'll have to work
according to the formula:

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

(a-b)^2 = a^2 - 2ab +
b^2

We'll analyze the
expression:

81y^6 +
180y^3*x^5+_=

We'll identify a^2 = 81y^6 => a = sqrt
81y^6

a = 9y^3

To calculate b,
we'll consider the second term of the square:

180y^3*x^5 =
2*a*b

180y^3*x^5 =
2*9y^3*b

We'll use the symmetric property and we'll divide
by -18y^3:

b =
180y^3*x^5/18y^3

b =
10x^5

Now, we'll complete the square by adding the amount
b^2.

b^2 = ( 10x^5)^2

b^2 =
100x^10

(a+b)^2 = (9y^3 +
10x^5)^2

The missing term in the quadratic expression is
100x^10:

(9y^3 + 10x^5)^2 = 81y^6 +
180y^3*x^5+100x^10

2) We notice that the
missing term is -2ab.

We'll identify a^2 = 16x^4/36
=> a = sqrt 16x^4/36 => a = 4x^2/6

16x^4/36 -
2*4x^2/6*b + 36x^2/16 = 0

To calculate 2ab, we'll consider
the 3rd term of the square:

36x^2/16 =
b^2

b = sqrt 36x^2/16

b =
-6x/4

2*4x^2/6*(-6x/4) =
-2x^3

The missing term in the quadratic expression is -2x^3
and the completed square will be:

(4x^2/6 -
6x/4)^2 = 16x^4/36-2x^3+36x^2/16