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Simplify: a - 1 + (1/a^2+a+1) / (a^3/a^3-1)Show solution and make sure that everything...

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spock5 | Student, Undergraduate | (Level 1) Honors

Posted May 3, 2012 at 5:12 PM via web

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Simplify: a - 1 + (1/a^2+a+1) / (a^3/a^3-1)

Show solution and make sure that everything is in its simplest form.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted May 3, 2012 at 5:20 PM (Answer #1)

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You need to bring to a common denominator the terms from numerator such that:

`(((a-1)(a^2 + a + 1) + 1)/(a^2 + a + 1))/((a^3)/(a^3-1))`

Notice that the special product `(a-1)(a^2 + a + 1)`   may be substituted by `a^3 - 1`  such that:

`((a^3 - 1 + 1)/(a^2 + a + 1))/((a^3)/(a^3-1))`

Reducing like terms yields:

`((a^3)/(a^2 + a + 1))/((a^3)/(a^3-1))`

`(1/(a^2 + a + 1))/(1/(a^3-1))`

Substituting`(a-1)(a^2 + a + 1)`  for`a^3-1`  yields:

`(1/(a^2 + a + 1))/(1/((a-1)(a^2 + a + 1)))`

Reducing like terms yields:

`(1/1)/(1/(a-1)) = a - 1`

Hence, simplifying the given fraction yields a - 1.

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