Solve for x fourth root of cube root of 9=3^x

$\sqrt[4]{\sqrt[3]{9}} = 3^{x}$

Step 1

Rewrite the equation as $3^{x} = \sqrt[4]{\sqrt[3]{9}}$ .

$3^{x} = \sqrt[4]{\sqrt[3]{9}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$3^{x} = \sqrt[4]{9^{\frac{1}{3}}}$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{9^{\frac{1}{3}}}$ as $9^{\frac{\frac{1}{3}}{4}}$ .

$3^{x} = 9^{\frac{\frac{1}{3}}{4}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$3^{x} = 9^{\frac{1}{3} \cdot \frac{1}{4}}$

Step 5

Multiply $\frac{1}{3} \cdot \frac{1}{4}$ .

Multiply $\frac{1}{3}$ by $\frac{1}{4}$ .

$3^{x} = 9^{\frac{1}{3 \cdot 4}}$

Multiply $3$ by $4$ .

$3^{x} = 9^{\frac{1}{12}}$

Step 6

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

$3^{x} = 3^{2 (\frac{1}{12})}$

Step 7

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

$x = 2 (\frac{1}{12})$

Step 8

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$x = 2 \frac{1}{2 (6)}$

Cancel the common factor.

$x = 2 \frac{1}{2 \cdot 6}$

Rewrite the expression.

$x = \frac{1}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{6}$

Decimal Form:

$x = 0.1\overline{6}$