How do you simplify #[(x+4)^3/4(x+2)^-2/3 - (x+2)^1/3(x+4)^-1/4]/ [(x+4)^3/4]^2#?

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Answer 1 · 2015-04-04T17:48:05

#4/(3(x+4)^3) *[1/(x+2)^2 - (x+2)/(x+4)^4]#

Given:
#(((x+4)^3)/4 * (x+2)^-2/3 - (x+2)/3 * (x+4)^-1/4)/((x+4)^3/4)^2#

Let #a = x+4# and #b=x+2#
#((a^3)/4 * b^-2/3 - b/3 * a^-1/4)/(a^3/4)^2#

When you divide by a fraction, you are multiplying the reciprocal:
#[(a^3)/4 * b^-2/3 - b/3 * a^-1/4] * (4/a^3)^2#

Change negative exponents to reciprocals with positive exponents:
#[(a^3)/4 * 1/(3b^2) - b/3 * 1/(4a)] * (4/a^3)^2#

#[a^3/(12b^2) - b/(12a)] (16/a^6)#

#(16a^3)/(12a^6b^2) - (16b)/(12a^7)#

#4/(3a^3b^2) - (4b)/(3a^7)#

Factor out #4/(3a^3)#:

#4/(3a^3) [1/b^2 - b/a^4]#

Substitute back in #x+4# and #x+2#
#4/(3(x+4)^3) * [1/(x+2)^2 - (x+2)/(x+4)^4]#

Hope that was helpful.