Solve for x 27^(9x+4)*3^(x-8)=3^(x+9)

$27^{9 x + 4} \cdot 3^{x - 8} = 3^{x + 9}$

Rewrite $27$ as $3^{3}$ .

${(3^{3})}^{9 x + 4} \cdot 3^{x - 8} = 3^{x + 9}$

Multiply the exponents in ${(3^{3})}^{9 x + 4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{3 (9 x + 4)} \cdot 3^{x - 8} = 3^{x + 9}$

Apply the distributive property.

$3^{3 (9 x) + 3 \cdot 4} \cdot 3^{x - 8} = 3^{x + 9}$

Multiply $9$ by $3$ .

$3^{27 x + 3 \cdot 4} \cdot 3^{x - 8} = 3^{x + 9}$

Multiply $3$ by $4$ .

$3^{27 x + 12} \cdot 3^{x - 8} = 3^{x + 9}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3^{27 x + 12 + x - 8} = 3^{x + 9}$

Add $27 x$ and $x$ .

$3^{28 x + 12 - 8} = 3^{x + 9}$

Subtract $8$ from $12$ .

$3^{28 x + 4} = 3^{x + 9}$

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

$28 x + 4 = x + 9$

Solve for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$28 x + 4 - x = 9$

Subtract $x$ from $28 x$ .

$27 x + 4 = 9$

Move all terms not containing $x$ to the right side of the equation.

Subtract $4$ from both sides of the equation.

$27 x = 9 - 4$

Subtract $4$ from $9$ .

$27 x = 5$

Divide each term in $27 x = 5$ by $27$ and simplify.

Divide each term in $27 x = 5$ by $27$ .

$\frac{27 x}{27} = \frac{5}{27}$

Simplify the left side.

Cancel the common factor of $27$ .

Cancel the common factor.

$\frac{27 x}{27} = \frac{5}{27}$

Divide $x$ by $1$ .

$x = \frac{5}{27}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{27}$

Decimal Form:

$x = 0.\overline{185}$