Algebra Examples

$36^{- 3 x + 3} = {(\frac{1}{216})}^{x + 1}$

Step 1

Apply the product rule to $\frac{1}{216}$ .

$36^{- 3 x + 3} = \frac{1^{x + 1}}{216^{x + 1}}$

Step 2

One to any power is one.

$36^{- 3 x + 3} = \frac{1}{216^{x + 1}}$

Step 3

Move $216^{x + 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$36^{- 3 x + 3} = 216^{- (x + 1)}$

Step 4

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

$6^{2 (- 3 x + 3)} = 6^{3 (- (x + 1))}$

Step 5

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

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

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 (- 3 x + 3)$ .

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Step 6.1.1

Rewrite.

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

Step 6.1.2

Simplify by adding zeros.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $- 3$ by $2$ .

$- 6 x + 2 \cdot 3 = 3 (- (x + 1))$

Step 6.1.4.2

Multiply $2$ by $3$ .

$- 6 x + 6 = 3 (- (x + 1))$

Step 6.2

Simplify $3 (- (x + 1))$ .

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Step 6.2.1

Apply the distributive property.

$- 6 x + 6 = 3 (- x - 1 \cdot 1)$

Step 6.2.2

Multiply $- 1$ by $1$ .

$- 6 x + 6 = 3 (- x - 1)$

Step 6.2.3

Apply the distributive property.

$- 6 x + 6 = 3 (- x) + 3 \cdot - 1$

Step 6.2.4

Multiply.

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Step 6.2.4.1

Multiply $- 1$ by $3$ .

$- 6 x + 6 = - 3 x + 3 \cdot - 1$

Step 6.2.4.2

Multiply $3$ by $- 1$ .

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

Step 6.3

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

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Step 6.3.1

Add $3 x$ to both sides of the equation.

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

Step 6.3.2

Add $- 6 x$ and $3 x$ .

$- 3 x + 6 = - 3$

Step 6.4

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

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Step 6.4.1

Subtract $6$ from both sides of the equation.

$- 3 x = - 3 - 6$

Step 6.4.2

Subtract $6$ from $- 3$ .

$- 3 x = - 9$

Step 6.5

Divide each term in $- 3 x = - 9$ by $- 3$ and simplify.

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Step 6.5.1

Divide each term in $- 3 x = - 9$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 9}{- 3}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 3$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 9}{- 3}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{- 3}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Divide $- 9$ by $- 3$ .

$x = 3$