Algebra Examples

$9^{2 x} \cdot 9^{1 - 3 x} = 27^{x + 2}$

Step 1

Multiply $9^{2 x}$ by $9^{1 - 3 x}$ by adding the exponents.

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Step 1.1

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

$9^{2 x + 1 - 3 x} = 27^{x + 2}$

Step 1.2

Subtract $3 x$ from $2 x$ .

$9^{- x + 1} = 27^{x + 2}$

Step 2

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

$3^{2 (- x + 1)} = 3^{3 (x + 2)}$

Step 3

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

$2 (- x + 1) = 3 (x + 2)$

Step 4

Solve for $x$ .

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Step 4.1

Simplify $2 (- x + 1)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + 2 (- x + 1) = 3 (x + 2)$

Step 4.1.2

Simplify by adding zeros.

$2 (- x + 1) = 3 (x + 2)$

Step 4.1.3

Apply the distributive property.

$2 (- x) + 2 \cdot 1 = 3 (x + 2)$

Step 4.1.4

Multiply.

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Step 4.1.4.1

Multiply $- 1$ by $2$ .

$- 2 x + 2 \cdot 1 = 3 (x + 2)$

Step 4.1.4.2

Multiply $2$ by $1$ .

$- 2 x + 2 = 3 (x + 2)$

Step 4.2

Simplify $3 (x + 2)$ .

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Step 4.2.1

Apply the distributive property.

$- 2 x + 2 = 3 x + 3 \cdot 2$

Step 4.2.2

Multiply $3$ by $2$ .

$- 2 x + 2 = 3 x + 6$

Step 4.3

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

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Step 4.3.1

Subtract $3 x$ from both sides of the equation.

$- 2 x + 2 - 3 x = 6$

Step 4.3.2

Subtract $3 x$ from $- 2 x$ .

$- 5 x + 2 = 6$

Step 4.4

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

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Step 4.4.1

Subtract $2$ from both sides of the equation.

$- 5 x = 6 - 2$

Step 4.4.2

Subtract $2$ from $6$ .

$- 5 x = 4$

Step 4.5

Divide each term in $- 5 x = 4$ by $- 5$ and simplify.

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Step 4.5.1

Divide each term in $- 5 x = 4$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{4}{- 5}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $- 5$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{4}{- 5}$

Step 4.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{- 5}$

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4}{5}$

Decimal Form:

$x = - 0.8$