Solve for x 3^(x^3)=9^x

$3^{x^{3}} = 9^{x}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$3^{x^{3}} = 3^{2 x}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x^{3} = 2 x$

Step 3

Solve for $x$ .

Subtract $2 x$ from both sides of the equation.

$x^{3} - 2 x = 0$

Factor $x$ out of $x^{3} - 2 x$ .

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - 2 x = 0$

Factor $x$ out of $- 2 x$ .

$x \cdot x^{2} + x \cdot - 2 = 0$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 2$ .

$x (x^{2} - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} - 2 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x^{2} - 2$ equal to $0$ and solve for $x$ .

Set $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Solve $x^{2} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$x^{2} = 2$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

The final solution is all the values that make $x (x^{2} - 2) = 0$ true.

$x = 0, \sqrt{2}, - \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 0, \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = 0, 1.41421356 \dots, - 1.41421356 \dots$

Solve for x 3^(x^3)=9^x

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