Algebra Examples

${(\frac{1}{3})}^{x} = 27^{x + 2}$

Step 1

Apply the product rule to $\frac{1}{3}$ .

$\frac{1^{x}}{3^{x}} = 27^{x + 2}$

Step 2

One to any power is one.

$\frac{1}{3^{x}} = 27^{x + 2}$

Step 3

Move $3^{x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 4

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

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

Step 5

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

$- x = 3 (x + 2)$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $3 (x + 2)$ .

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Multiply $3$ by $2$ .

$- x = 3 x + 6$

Step 6.2

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

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Step 6.2.1

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

$- x - 3 x = 6$

Step 6.2.2

Subtract $3 x$ from $- x$ .

$- 4 x = 6$

Step 6.3

Divide each term in $- 4 x = 6$ by $- 4$ and simplify.

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Step 6.3.1

Divide each term in $- 4 x = 6$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{6}{- 4}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $- 4$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{6}{- 4}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{- 4}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Cancel the common factor of $6$ and $- 4$ .

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Step 6.3.3.1.1

Factor $2$ out of $6$ .

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

Step 6.3.3.1.2

Cancel the common factors.

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Step 6.3.3.1.2.1

Factor $2$ out of $- 4$ .

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

Step 6.3.3.1.2.2

Cancel the common factor.

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

Step 6.3.3.1.2.3

Rewrite the expression.

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

Step 6.3.3.2

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.5$

Mixed Number Form:

$x = - 1 \frac{1}{2}$