Solve for b 81*9^(-2b-2)=27

$81 \cdot 9^{- 2 b - 2} = 27$

Rewrite $81$ as $3^{4}$ .

$3^{4} \cdot 9^{- 2 b - 2} = 27$

Rewrite $9$ as $3^{2}$ .

$3^{4} \cdot {(3^{2})}^{- 2 b - 2} = 27$

Multiply the exponents in ${(3^{2})}^{- 2 b - 2}$ .

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

$3^{4} \cdot 3^{2 (- 2 b - 2)} = 27$

Apply the distributive property.

$3^{4} \cdot 3^{2 (- 2 b) + 2 \cdot - 2} = 27$

Multiply $- 2$ by $2$ .

$3^{4} \cdot 3^{- 4 b + 2 \cdot - 2} = 27$

Multiply $2$ by $- 2$ .

$3^{4} \cdot 3^{- 4 b - 4} = 27$

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

$3^{4 - 4 b - 4} = 27$

Subtract $4$ from $4$ .

$3^{- 4 b + 0} = 27$

Add $- 4 b$ and $0$ .

$3^{- 4 b} = 27$

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

$3^{- 4 b} = 3^{3}$

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

$- 4 b = 3$

Divide each term in $- 4 b = 3$ by $- 4$ and simplify.

Divide each term in $- 4 b = 3$ by $- 4$ .

$\frac{- 4 b}{- 4} = \frac{3}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 b}{- 4} = \frac{3}{- 4}$

Divide $b$ by $1$ .

$b = \frac{3}{- 4}$

Simplify the right side.

Move the negative in front of the fraction.

$b = - \frac{3}{4}$

The result can be shown in multiple forms.

Exact Form:

$b = - \frac{3}{4}$

Decimal Form:

$b = - 0.75$