6n5x^3÷18nx^7
12m^8y^6÷-9my^4

(-n^4 - 4 m^7 x^4 y^2)/(3 x^4)

Step-by-step explanation:

Simplify the following:

(-6 n^5 x^3)/(18 n x^7) + (12 m^8 y^6)/(-9 m y^4)

The gcd of -6 and 18 is 6, so (-6 n^5 x^3)/(18 n x^7) = ((6 (-1)) n^5 x^3)/((6×3) n x^7) = 6/6×(-n^5 x^3)/(3 n x^7) = (-n^5 x^3)/(3 n x^7):

(-1 n^5 x^3)/(3 n x^7) + (12 m^8 y^6)/(-9 m y^4)

Combine powers. (-n^5 x^3)/(3 n x^7) = (n^(5 - 1) x^(3 - 7) (-1))/3:

-(n^(5 - 1) x^(3 - 7))/3 + (12 m^8 y^6)/(-9 m y^4)

5 - 1 = 4:

-(n^4 x^(3 - 7))/3 + (12 m^8 y^6)/(-9 m y^4)

3 - 7 = -4:

-(n^4 x^(-4))/3 + (12 m^8 y^6)/(-9 m y^4)

The gcd of 12 and -9 is 3, so (12 m^8 y^6)/(-9 m y^4) = ((3×4) m^8 y^6)/((3 (-3)) m y^4) = 3/3×(4 m^8 y^6)/(-3 m y^4) = (4 m^8 y^6)/(-3 m y^4):

(4 m^8 y^6)/(-3 m y^4) - (n^4)/(3 x^4)

Combine powers. (4 m^8 y^6)/(-3 m y^4) = (4 m^(8 - 1) y^(6 - 4))/(3 (-1)):

-(n^4)/(3 x^4) + (4 m^(8 - 1) y^(6 - 4))/(-3)

8 - 1 = 7:

-(n^4)/(3 x^4) + (4 m^7 y^(6 - 4))/(-3)

6 - 4 = 2:

-(n^4)/(3 x^4) + (4 m^7 y^2)/(-3)

Multiply numerator and denominator of (4 m^7 y^2)/(-3) by -1:

-4/3 m^7 y^2 - (n^4)/(3 x^4)

Put each term in -(n^4)/(x^4×3) - (4 m^7 y^2)/3 over the common denominator 3 x^4: -(n^4)/(x^4×3) - (4 m^7 y^2)/3 = (-n^4)/(3 x^4) - (4 m^7 x^4 y^2)/(3 x^4):

-n^4/(3 x^4) - (4 m^7 x^4 y^2)/(3 x^4)

-n^4/(3 x^4) - (4 m^7 x^4 y^2)/(3 x^4) = (-n^4 - 4 m^7 x^4 y^2)/(3 x^4):

(-n^4 - 4 m^7 x^4 y^2)/(3 x^4)

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step-by-step explanation: