and simplify the given expressions,

1. (11x² + 3x + 19) + (6x² - 18x - 8)

2. (13x² + 6xy -12y²) - (9x² - 4xy)

3. -2xy(3xycubed - 5xy + 7y²)

4. (11 + √3) (4 - √11)

and simplify the given expressions,

1. (11x² + 3x + 19) + (6x² - 18x - 8)

2. (13x² + 6xy -12y²) - (9x² - 4xy)

3. -2xy(3xycubed - 5xy + 7y²)

4. (11 + √3) (4 - √11)

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√144 = 12 which is a positive integer. Thus one set it belongs to is N (natural numbers). It also belongs to Z (integers), Q (rational numbers), R (real numbers), and C (complex numbers) since N is a subset of all of these sets.

For the expressions:

1. Combine like terms

(11x^{2} + 3x + 19) + (6x^{2} - 18x - 8) = (11 + 6)x^{2} + (3 - 18)x + (19 - 8)

= 17x^{2} - 15x + 11.

2. Same idea as 1.

(13x^{2} + 6xy - 12y^{2}) - (9x^{2} - 4xy) = (13 - 9)x^{2} + (6 - (-4))xy - 12y^{2}

= 4x^{2} + 10xy - 12y^{2}

3. Use the distributive property a(b+c), and the law of exponents x^{m}x^{n
}= x^{m+n}

-2xy(3xy^{3} - 5xy + 7y^{2}) = -6x^{2}y^{4} + 10x^{2}y^{2} - 14xy^{5}

4. Use distributive property. Also apply property of square roots: √(xy) = (√x)(√y)

(11+√3)(4 - √11) = 11(4 - √11) + (√3)(4 - √11) = 44 - 11√11 + 4√3 - √33.