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If x^3x = 3x^4x, then find x.

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nasirjam | Student, Grade 9 | Honors

Posted September 21, 2012 at 8:53 AM via web

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If x^3x = 3x^4x, then find x.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted September 21, 2012 at 2:03 PM (Answer #1)

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You should move the terms to one side such that:

`x^(3x)- 3x^(4x) = 0`

You need to factor out `x^(3x) `  such that:

`x^(3x)(1 - 3x^(4x-3x)) = 0 => x^(3x)(1 - 3x^x) = 0 `

Since `x^(3x)!=0` , then only `1 - 3x^x = 0 => 3x^x = 1`

You need to divide by 3 both sides such that:

`x^x = 1/3`

You need to sketch the graphs of the function `y=x^x`  and the graph of the function `y=1/3`  such that:

Notice that the graphs do not intersect, hence, the given equation has no solution.

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samhouston | Middle School Teacher | (Level 1) Associate Educator

Posted September 21, 2012 at 11:43 AM (Answer #3)

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x^3x = 3x^4x

Divide both sides by x^3x

0 = (3x^4x) / (x^3x)

According to laws of exponents, when dividing powers, the exponents are subtracted.

(x^4x) / (x^3x) = x^(4x-3x) = x^x

Now we have the equation

0 = 3x^x

Divide both sides by 3.

0 = x^x

In order for a power to equal 0, the base would have to be 0 and the exponent would have to be a positive number.  Since both the base and the exponent are x (and therefore the same number), this is impossible. 

There is no solution.

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