What is the simplified Rational expression of this; m^6 - n^6 over m^4 - n^4?
928756
m^6-n^6=(m^3+n^3)(m^3-n^3)\n(m^2-n^2)(m^4+m^2n^2+n^4)\nm^4-n^4=(m^2+n^2)(m^2-n^2)\nplugging in\nm^6-n^6/m^4-n^4\n=(m^4+m^2n^2+n^4)/(m^2+n^2)
766061
p^9-q^9/p^6-q^6\n=(p^3-q^3)(p^6+p^3q^3+q^6)/(p^3+q^3)(p^3-q^3)\n=(p^6+p^3q^3+q^6)/(p^3+q^3)
798514
do you mean 6x([1/2]x^2)?\nif so 3x^3\nor 6x(1/(2x^2))?\nthen its 3x^-1
966462
(6x^2)/(4x^2 y-12xy) * (x^2 + x - 12)/ (3x^2 + 12x)\n\nFirst, I would attempt to simplify if possible:\n2x(3x)/x(2xy-6y) * (x^2 + x - 12)/ (3x^2 + 12x )\n(3x) / (2xy - 6y) * (x^2 + x - 12)/ (3x^2 + 12x)\n\nRewrite to see if we can cross cancel:\n\n(3x)           (x^2 + x - 12)\n---------  *   -----------------\n(2xy-6y)    (3x)(x + 4x)\n\n1               (x^2 + x - 12)\n---------- *  -----------------\n(2xy-6y)     (x+4x)\n\nUnfoil the (x^2 + x -12 to see if we can further simplify\n\n(x^2 + x - 12) = (x -3)(x+4) so you will have:\n\n1             (x-3)(x+4)\n-------  *  --------------\n2y(x-3)      (x+4)\n\nBy simplification:\n\n1           (x-3)\n------   * ----------\n2y(x-3)     1\n\nBy Cross Cancel:\n1/2y
712942
x^6 - 6x^5 + 15x^4 - 30x^3 + 15x^2 - 6x + 1 = 0\nx^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 - 10x^3 = 0\nSo (x - 1)^6 - 10x^3 = 0\nSo ((x - 1)²)³ - (³√10x)³ = 0\n\nSo ((x - 1)² - ³√10x)((x - 1)^4 + ³√10x((x - 1)² + ³√100x²) = 0\n\nSo ((x - 1)^4 + ³√10x((x - 1)² + ³√100x²) = 0 N0 real roots (Min value near x = 0.2 just more than 0.87)\n\nOR (x - 1)² - ³√10x = 0 \nie x² - (2 + ³√10)x + 1 = 0\nie x = ½[(2 + ³√10) ± √(4 + 4³√10 + ³√100 - 4)\n= ½[(2 + ³√10) ± √(4³√10 + ³√100)]\n\nSOLUTION\nx = ½[(2 + ³√10) ± √(4³√10 + ³√100)]
1016236
(1/(z^2 - 3z - 4)) + ((3z)/(z^2 - 6z - 7))\n(1/((z - 4)(z + 1))) + ((3z)/((z - 7)(z + 1)))\n\nMultiply everything by (z - 4)(z + 1)(z - 7)\n\n((z - 7) + (3z(z - 4)))/((z - 4)(z + 1)(z - 7))\n\n(z - 7 + 3z^2 - 12z)/((z - 4)(z + 1)(z - 7))\n\n(3z^2 - 11z - 7)/((z - 4)(z + 1)(z - 7))\n\nANS : a.) (3z^2 - 11z - 7)/((z - 4)(z + 1)(z - 7))
727905
(a + b)^6 - (a - b)^6\n\nNow, the previous answerers chose to factor the difference of 2 squares first, but I prefer to expand first.\n= (a^6 + 6a^5 b + 15a^4 b² + 20a³b³ + 15a²b^4 + 6ab^5 + b^6) - (a^6 - 6a^5 b + 15a^4 b² - 20a³b³ + 15a²b^4 - 6ab^5 + b^6)\n\nNow, we distribute the negative sign:\n= a^6 + 6a^5 b + 15a^4 b² + 20a³b³ + 15a²b^4 + 6ab^5 + b^6 - a^6 + 6a^5 b - 15a^4 b² + 20a³b³ - 15a²b^4 + 6ab^5 - b^6\n\nWe combine like terms\n= 12a^5 b + 40a³b³ + 12ab^5\n\nWe factor out 4ab\n= 4ab(3a^4 + 10a²b² + 3b^4)\n\nWe factor out the biquadratic\n= 4ab(3a² + b²)(a² + 3b²)\n\nTherefore, the completely factored form is\n(a + b)^6 - (a - b)^6 = 4ab(3a² + b²)(a² + 3b²)\n\n^_^
246754
6y^2, this leaves 6y^2(1-2y+6y^2)
685531
[(2x^3+2x^2+3x+3)/(x^2-9)]*\n[(x^2+2x-15)/(4x^2+6)]\n\nFactor each expression individually.\n\n(2x^3+2x^2+3x+3) = (x+1)(2x^2+3)\n\n(x^2-9) = (x+3)(x-3)\n\n(x^2+2x-15) = (x+5)(x-3)\n\n(4x^2+6) = 2(2x^2+3)\n\nSubstituting these in, our expression becomes\n{[(x+1)(2x^2+3)]/[(x+3)(x-3)]}\n*{[(x+5)(x-3)]/[2(2x^2+3)]}\n\nCombining these into one larger expression, we get\n[(x+1)(2x^2+3)(x+5)(x-3)]/\n[2(x+3)(x-3)(2x^2+3)]\n\nCancel the quantities common to the numerator and denominator and we get our final answer\n[(x+1)(x+5)]/[2(x+3)]\n\nGood luck.
810486
Hi Arman! With no L’Hopital again? \n((x^1/4)-(x^1/3))/((x^1/2)-1) for x->1, mind! x>0\nLet z=x^1/12, then (z^3-z^4)/(z^6-1) =\n= z^3*(1-z) / {(z^3-1)*(z^3+1)} = \n= z^3*(1-z) / {(z-1)*(z^2+z+1)*(z^3+1)} = \n= -z^3/{(z^2+z+1)*(z^3+1)} = \n= /for z->1/ = -1/(3*2)=-1/6
443446
S (x^2-4x) / (x^3-6x^2) dx\n\nlet u = x^3-6x^2\nthen du = 3x^2-12x\n\nS (x^2-4x) / (x^3-6x^2) dx = (1/3) du/u\n= (1/3) ln u + C\n= (1/3) ln (x^3-6x^2) + C\n\nAnswer B.
817461
a^4 - b^4\n\n= (a^2 + b^2)(a^2 - b^2)\n\n= (a^2 + b^2)(a + b)(a - b)
725839
I think it is this:\n\nFirst multiply through the groups..\nso you get:\n6n^7-4n^6-16+4n^6+2n^7+15\n\nThen gather like terms\n\n=8n^7-1\n\n(The -4n^6 and the +4n^6 cancel each other out)\n\nI don't think you can go further than that.\n\nGood luck.
757614
4 (x^2 + 3X -4)\n4(x+4)(X-1)
714160
x^2+6x+8/x^2-4-21 / x^2-4/x^2-7x\n=x^2 + 6x +(8/x^2) - 4 - (21/x^2) - (4/x^2) - 7x\n=x^2 + 6x -4 -7x +(8/x^2) - (21/x^2) - (4/x^2) \n=x^2 + 6x -4 -7x +[8-21-4]/x^2\n=x^2 + 6x -4 -7x -17/x^2\n=(x^2)*[x^2 - x -4  -17]/x^2\n=(x^2)*(x^2) - (x^2)*(x) -4*(x^2)  -17]/x^2\n=(x^4) - (x^3) - 4(x^2)x -17]/x^2
896236
First expand the brackets, collect like terms and simplify.\n\n(-2x^2 + 5x - 1) - (3x^2 - 4x - 6)\n= -2x^2 + 5x - 1 - 3x^2 + 4x + 6\n= (-2 - 3)x^2 + (5 + 4)x + (-1 + 6)\n= -5x^2 + 9x + 5
352749
By the chain rule:\n[e^(3x^(2)-7x)]*(6x-7)