Precalculus Examples

$27^{1 - x} = {(\frac{1}{9})}^{2 - x}$

Step 1

Apply the product rule to $\frac{1}{9}$ .

$27^{1 - x} = \frac{1^{2 - x}}{9^{2 - x}}$

Step 2

One to any power is one.

$27^{1 - x} = \frac{1}{9^{2 - x}}$

Step 3

Move $9^{2 - x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$27^{1 - x} = 9^{- (2 - x)}$

Step 4

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

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

Step 5

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

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

Step 6

Solve for $x$ .

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Step 6.1

Simplify $3 (1 - x)$ .

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Step 6.1.1

Rewrite.

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

Step 6.1.2

Simplify by adding zeros.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $3$ by $1$ .

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

Step 6.1.4.2

Multiply $- 1$ by $3$ .

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

Step 6.2

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

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Multiply $- 1$ by $2$ .

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

Step 6.2.3

Multiply $- - x$ .

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Step 6.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2.3.2

Multiply $x$ by $1$ .

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

Step 6.2.4

Apply the distributive property.

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

Step 6.2.5

Multiply $2$ by $- 2$ .

$3 - 3 x = - 4 + 2 x$

Step 6.3

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

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Step 6.3.1

Subtract $2 x$ from both sides of the equation.

$3 - 3 x - 2 x = - 4$

Step 6.3.2

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

$3 - 5 x = - 4$

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 $3$ from both sides of the equation.

$- 5 x = - 4 - 3$

Step 6.4.2

Subtract $3$ from $- 4$ .

$- 5 x = - 7$

Step 6.5

Divide each term in $- 5 x = - 7$ by $- 5$ and simplify.

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Step 6.5.1

Divide each term in $- 5 x = - 7$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 7}{- 5}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 5$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 7}{- 5}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{- 5}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Dividing two negative values results in a positive value.

$x = \frac{7}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{5}$

Decimal Form:

$x = 1.4$

Mixed Number Form:

$x = 1 \frac{2}{5}$