Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Multiply $- 2$ by $2$ .

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

Step 3.4

Multiply $2$ by $- 2$ .

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

Step 4

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

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

Step 5

Subtract $4$ from $4$ .

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

Step 6

Add $- 4 x$ and $0$ .

$3^{- 4 x} = 27$

Step 7

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

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

Step 8

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

$- 4 x = 3$

Step 9

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

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Step 9.1

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

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

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 4$ .

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Step 9.2.1.1

Cancel the common factor.

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

Step 9.2.1.2

Divide $x$ by $1$ .

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

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.75$