Solve for x square root of 6=216^x

$\sqrt{6} = 216^{x}$

Step 1

Rewrite the equation as $216^{x} = \sqrt{6}$ .

$216^{x} = \sqrt{6}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$216^{x} = 6^{\frac{1}{2}}$

Step 3

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

$6^{3 x} = 6^{\frac{1}{2}}$

Step 4

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

$3 x = \frac{1}{2}$

Step 5

Divide each term in $3 x = \frac{1}{2}$ by $3$ and simplify.

Divide each term in $3 x = \frac{1}{2}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{1}{2}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{1}{2}}{3}$

Divide $x$ by $1$ .

$x = \frac{\frac{1}{2}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{2} \cdot \frac{1}{3}$

Multiply $\frac{1}{2} \cdot \frac{1}{3}$ .

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$x = \frac{1}{2 \cdot 3}$

Multiply $2$ by $3$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.1\overline{6}$