Given the law of composition x*y=xy+4x+4y+12, solve the following equation:x*x*x*x

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Since you completed the right side of the equation a bit later, the equation that is solved is x*x*x*x = 0, hence, solving the equation x*x*x*x = 12 yields:

`(x+4)^4 - 4 = 12 => (x+4)^4 = 16 => x+4 = root(4)(16) => x+4 =+-2`

`x + 4 = 2 => x = -2`

`x + 4 = -2 => x = -6`

Hence, evaluating the solution to the equation x*x*x*x = 12 yields x = -2 and x = -6.

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Notice that you may write the law of composition such that:

`x*y = (x+4)(y+4)-4`

Resoning by analogy yields:

`x*x = (x+4)(x+4)-4 => x*x = (x+4)^2 - 4`

Using `x*x = (x+4)^2 - 4` , you may evaluate `(x*x)*(x*x)`  such that:

`(x*x)*(x*x)= ((x+4)^2 - 4 + 4)((x+4)^2 - 4 + 4) - 4`

`(x*x)*(x*x) = ((x+4)^2)((x+4)^2) - 4 => (x*x)*(x*x) = (x+4)^4 - 4`

You need to solve for x the equation `(x*x)*(x*x) = 0`  such that:

`(x+4)^4 - 4 = 0`

You should convert the difference of squares into a product such that:

`((x+4)^2 - 2)((x+4)^2 + 2) = 0`

You should solve the following equations such that:

`(x+4)^2 - 2 = 0 => (x+4)^2= 2 => x+4 = +-sqrt2 => x_(1,2) = -4+-sqrt2`

`(x+4)^2+ 2 = 0 => (x+4)^2 = -2 => x_(3,4) = -4+-i*sqrt2`

Hence, evaluating the solutions to the given equation yields `x_(1,2) = -4+-sqrt2`  and `x_(3,4) = -4+-i*sqrt2` .

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emanuel- | Student, College Freshman | (Level 2) eNoter

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x*x*x*x = 12

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