(27x^3-64)/(9x^2-16)Simplify .

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To evaluate the expression we'll use factorization. We notice that the numerator is a difference of cubes:

27x^3-64 = (3x)^3 - (4)^3

We'll apply the formula:

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

We'll put a = 3x and b = 4

(3x)^3 - (4)^3 = (3x-4)(9x^2 + 12x + 16)

We also notice that the denominator is a difference of squares:

9x^2-16 = (3x)^2 - 4^2

We'll apply the formula:

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

(3x)^2 - 4^2 = (3x-4)(3x+4)

We'll substitute the differences by their products:

[(27x^3-64)/(9x^2-16)] = (3x-4)(9x^2 + 12x + 16)/(3x-4)(3x+4)

We'll simplify by the common factor (3x-4):

[(27x^3-64)/(9x^2-16)] =  [(9x^2 + 12x + 16)/(3x+4)]

We can also combine the terms 12x + 16 and factorize them by 4;

[(27x^3-64)/(9x^2-16)]  = 9x^2/(3x+4) + (12x + 16)/(3x+4)

[(27x^3-64)/(9x^2-16)]  = 9x^2/(3x+4) + 4(3x + 4)/(3x+4)

We'll simplify the last ratio by (3x+4) and we'll get:

[(27x^3-64)/(9x^2-16)]  = 9x^2/(3x+4) + 4

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