Let's determine which of the given expressions is a perfect square trinomial.
A perfect square trinomial takes the form \((a + b)^2\) or \((a - b)^2\), which expands to \(a^2 + 2ab + b^2\) for the addition form or \(a^2 - 2ab + b^2\) for the subtraction form.
Let's analyze each given expression:
1. \(50y^2 - 4x^2\)
This expression can be factored as a difference of squares:
[tex]\[50y^2 - 4x^2 = (5\sqrt{2}y)^2 - (2x)^2 = (5\sqrt{2}y - 2x)(5\sqrt{2}y + 2x)\][/tex]
This is not a perfect square trinomial.
2. \(100 - 36x^2y^2\)
This expression can also be factored as a difference of squares:
[tex]\[100 - 36x^2y^2 = (10)^2 - (6xy)^2 = (10 - 6xy)(10 + 6xy)\][/tex]
This too is not a perfect square trinomial.
3. \(16x^2 + 24xy + 9y^2\)
Let's factor this expression:
[tex]\[16x^2 + 24xy + 9y^2 = (4x + 3y)^2\][/tex]
[tex]\[
\text{Check the middle term: } 2ab = 2(4x)(3y) = 24xy
\][/tex]
This expression matches the form \(a^2 + 2ab + b^2\), where \(a = 4x\) and \(b = 3y\).
Thus, \(16x^2 + 24xy + 9y^2\) is a perfect square trinomial.
4. \(49x^2 - 70xy + 10y^2\)
Consider whether this can be expressed as a square trinomial:
[tex]\[
(7x)^2 = 49x^2 \quad \text{and} \quad (c)^2 = 10y^2
\][/tex]
However, the middle term \(-70xy\) does not satisfy \(2ab \neq -70xy\).
Thus, \(49x^2 - 70xy + 10y^2\) is not a perfect square trinomial.
Based on the detailed analysis, the expression that is a perfect square trinomial is:
[tex]\[
16x^2 + 24xy + 9y^2
\][/tex]
Therefore, the index of the perfect square trinomial is 3.
