Answer:
(3/5a^3b -5/6ab^3)^2
Step-by-step explanation:
Your trinomial
9/25a^6b^2 - a^4b^4 + 25/36a^2b^6
has three terms, as it must if it is to be the square of a binomial. The terms all have degree 8, so each of the terms of its square root will have degree 8/2 = 4.
The term with the highest degree of the variable "a" is the first one ...
9/25a^6b^2
All parts of this term are perfect squares:
- 9/25 = (3/5)^2
- a^6 = (a^3)^2
- b^2 = (b)^2
so, this term can be written as the square ...
(3/5a^3b)^2
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The term with the highest degree of the variable "b" is the last one ...
25/36a^2b^6
All parts of this term are perfect squares:
- 25/36 = (5/6)^2
- a^2 = (a)^2
- b^6 = (b^3)^2
so, this term can be written as the square ...
(5/6ab^3)^2
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If the given trinomial is truly the square of a binomial, then the remaining term, the middle one, will be twice the product of the terms of the binomial. Here that means we want to make sure that ...
a^4b^4 = 2 · (3/5a^3b) · (5/6ab^3)
= 2 · 3/5 · 5/6 · a^4b^4
= a^4b^4 . . . . . . as required
So, we know the two terms of the binomial, 3/5a^3b and 5/6ab^3, and we know the sign of the middle term of the given trinomial is negative. That negative sign means the sign in our binomial must be negative. Then the expression we're asked to write is ...
9/25a^6b^2 - a^4b^4 + 25/36a^2b^6 = (3/5a^3b -5/6ab^3)^2
