# Thread: Solve Equation w/ Rational Exponent.

1. ## Solve Equation w/ Rational Exponent.

I am lost. The Algebra II book I use doesn't explain concepts very well. Does anyone know how to solve this problem as well as clearly explain their chosen steps?

$\displaystyle x^{\frac{1}{2}} -5x^{\frac{1}{4}} +6=0$

2. Hello, Mulya66!

I hope there's a typo in the problem.
. . Otherwise, I can't solve it.
. . . . . . . . . . . . . . . . . . . . .

Is it supposed to be: .$\displaystyle x^{\frac{2}{3}} -5x^{\frac{1}{3}} + 6 \:=\:0$

Then it factors: .$\displaystyle \left(x^{\frac{1}{3}} - 2\right)\left(x^{\frac{1}{3}} - 3\right) \:=\:0$

And we have: .$\displaystyle \begin{Bmatrix}x^{\frac{1}{3}} - 2 \:=\:0\quad\Rightarrow\quad x^{\frac{1}{3}}\:=\:2\quad\Rightarrow\quad x\,=\,8 \\ x^{\frac{1}{3}} - 3\:=\:0\quad\Rightarrow\quad x^{\frac{1}{3}}\:=\:3\quad\Rightarrow\quad x\,=\,27 \end{Bmatrix}$

3. Oops! It's all a typo! It's really $\displaystyle x^{\frac{1}{2}} -5x^{\frac{1}{4}} + 6 \:=\:0$

(Well, that just shows how tired I am...)

Originally Posted by Soroban
Hello, Mulya66!

I hope there's a typo in the problem.
. . Otherwise, I can't solve it.
. . . . . . . . . . . . . . . . . . . . .

Is it supposed to be: .$\displaystyle x^{\frac{2}{3}} -5x^{\frac{1}{4}} + 6 \:=\:0$

Then it factors: .$\displaystyle \left(x^{\frac{1}{3}} - 2\right)\left(x^{\frac{1}{3}} - 3\right) \:=\:0$

And we have: .$\displaystyle \begin{Bmatrix}x^{\frac{1}{3}} - 2 \:=\:0\quad\Rightarrow\quad x^{\frac{1}{3}}\:=\:2\quad\Rightarrow\quad x\,=\,8 \\ x^{\frac{1}{3}} - 3\:=\:0\quad\Rightarrow\quad x^{\frac{1}{3}}\:=\:3\quad\Rightarrow\quad x\,=\,27 \end{Bmatrix}$

4. Hello, Mulya66!

$\displaystyle x^{\frac{1}{2}} -5x^{\frac{1}{4}} + 6 \:=\:0$

Use the same technique . . .

It factors: .$\displaystyle \left(x^{\frac{1}{4}} - 2\right)\left(x^{\frac{1}{4}} - 3\right) \:=\:0$

Can you finish it now?

5. Originally Posted by Soroban

$\displaystyle \left(x^{\frac{1}{4}} - 2\right)\left(x^{\frac{1}{4}} - 3\right) \:=\:0$
$\displaystyle \begin{Bmatrix}x^{\frac{1}{4}} - 2 \:=\:0\quad\Rightarrow\quad x^{\frac{1}{4}}\:=\:2\quad\Rightarrow\quad x\,=\,16 \\ x^{\frac{1}{4}} - 3\:=\:0\quad\Rightarrow\quad x^{\frac{1}{4}}\:=\:3\quad\Rightarrow\quad x\,=\,81 \end{Bmatrix}$

Thanks! It's very easy now.