What is expand (54e^x(-e^x+2))/((3e^x+6)^3) ?

The solution to expand (54e^x(-e^x+2))/((3e^x+6)^3) is (-2e^{2x}+4e^x)/(e^{3x)+6e^{2x}+12e^x+8}

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expand (54e^x(-e^x+2))/((3e^x+6)^3)

expand

\frac{54 e ^{x} ( - e ^{x} + 2 )}{( 3 e ^{x} + 6 ) ^{3}}

\frac{- 2 e ^{2 x} + 4 e ^{x}}{e ^{3 x} + 6 e ^{2 x} + 12 e ^{x} + 8}

Solution steps

\frac{54 e ^{x} ( - e ^{x} + 2 )}{( 3 e ^{x} + 6 ) ^{3}}

(3 e^{x} + 6)^{3} = 27 e^{3 x} + 162 e^{2 x} + 324 e^{x} + 216

= \frac{54 e ^{x} ( - e ^{x} + 2 )}{27 e ^{3 x} + 162 e ^{2 x} + 324 e ^{x} + 216}

Factor

27 e^{3 x} + 162 e^{2 x} + 324 e^{x} + 216 : 27 (e^{3 x} + 6 e^{2 x} + 12 e^{x} + 8)

= \frac{54 e ^{x} ( - e ^{x} + 2 )}{27 ( e ^{3 x} + 6 e ^{2 x} + 12 e ^{x} + 8 )}

Divide the numbers:

\frac{54}{27} = 2

= \frac{2 e ^{x} ( - e ^{x} + 2 )}{( e ^{3 x} + 6 e ^{2 x} + 12 e ^{x} + 8 )}

Remove parentheses:

(a) = a

= \frac{2 e ^{x} ( - e ^{x} + 2 )}{e ^{3 x} + 6 e ^{2 x} + 12 e ^{x} + 8}

Expand

2 e^{x} (- e^{x} + 2) : - 2 e^{2 x} + 4 e^{x}

= \frac{- 2 e ^{2 x} + 4 e ^{x}}{e ^{3 x} + 6 e ^{2 x} + 12 e ^{x} + 8}

\frac{2 ( 1 - 13 )}{3}

- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x}))

, (8 x - 5) (2 x^{2} - 5 x - 6)

lo g_{2} (4)

What is expand (54e^x(-e^x+2))/((3e^x+6)^3) ?
The solution to expand (54e^x(-e^x+2))/((3e^x+6)^3) is (-2e^{2x}+4e^x)/(e^{3x)+6e^{2x}+12e^x+8}