Solve for x square root of 27^x=81

$\sqrt{27^{x}} = 81$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{27^{x}}$ as $27^{\frac{x}{2}}$ .

$27^{\frac{x}{2}} = 81$

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

$3^{3 \frac{x}{2}} = 3^{4}$

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

$3 \frac{x}{2} = 4$

Solve for $x$ .

Combine $3$ and $\frac{x}{2}$ .

$\frac{3 x}{2} = 4$

Divide each term in $\frac{3 x}{2} = 4$ by $3$ and simplify.

Divide each term in $\frac{3 x}{2} = 4$ by $3$ .

$\frac{\frac{3 x}{2}}{3} = \frac{4}{3}$

Simplify the left side.

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 x}{2} \cdot \frac{1}{3} = \frac{4}{3}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$\frac{3 (x)}{2} \cdot \frac{1}{3} = \frac{4}{3}$

Cancel the common factor.

$\frac{3 x}{2} \cdot \frac{1}{3} = \frac{4}{3}$

Rewrite the expression.

$\frac{x}{2} = \frac{4}{3}$

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (\frac{4}{3})$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{x}{2} = 2 (\frac{4}{3})$

Rewrite the expression.

$x = 2 (\frac{4}{3})$

Simplify the right side.

Multiply $2 (\frac{4}{3})$ .

Combine $2$ and $\frac{4}{3}$ .

$x = \frac{2 \cdot 4}{3}$

Multiply $2$ by $4$ .

$x = \frac{8}{3}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{3}$

Decimal Form:

$x = 2.\overline{6}$

Mixed Number Form:

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