Algebra Examples

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

Step 1

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

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

Step 2

One to any power is one.

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

Step 3

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

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

Step 4

Rewrite $27$ as $3^{3}$ .

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

Step 5

Multiply the exponents in ${(3^{3})}^{x + 1}$ .

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Step 5.1

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

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $3$ by $1$ .

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

Step 6

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

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

Step 7

Add $- x$ and $3 x$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Solve for $x$ .

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Step 10.1

Simplify $2 (3 - 2 x)$ .

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Step 10.1.1

Rewrite.

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

Step 10.1.2

Simplify by adding zeros.

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

Step 10.1.3

Apply the distributive property.

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

Step 10.1.4

Multiply.

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Step 10.1.4.1

Multiply $2$ by $3$ .

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

Step 10.1.4.2

Multiply $- 2$ by $2$ .

$6 - 4 x = 2 x + 3$

Step 10.2

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

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Step 10.2.1

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

$6 - 4 x - 2 x = 3$

Step 10.2.2

Subtract $2 x$ from $- 4 x$ .

$6 - 6 x = 3$

Step 10.3

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

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Step 10.3.1

Subtract $6$ from both sides of the equation.

$- 6 x = 3 - 6$

Step 10.3.2

Subtract $6$ from $3$ .

$- 6 x = - 3$

Step 10.4

Divide each term in $- 6 x = - 3$ by $- 6$ and simplify.

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Step 10.4.1

Divide each term in $- 6 x = - 3$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 3}{- 6}$

Step 10.4.2

Simplify the left side.

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Step 10.4.2.1

Cancel the common factor of $- 6$ .

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Step 10.4.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 3}{- 6}$

Step 10.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 6}$

Step 10.4.3

Simplify the right side.

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Step 10.4.3.1

Cancel the common factor of $- 3$ and $- 6$ .

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Step 10.4.3.1.1

Factor $- 3$ out of $- 3$ .

$x = \frac{- 3 (1)}{- 6}$

Step 10.4.3.1.2

Cancel the common factors.

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Step 10.4.3.1.2.1

Factor $- 3$ out of $- 6$ .

$x = \frac{- 3 \cdot 1}{- 3 \cdot 2}$

Step 10.4.3.1.2.2

Cancel the common factor.

$x = \frac{- 3 \cdot 1}{- 3 \cdot 2}$

Step 10.4.3.1.2.3

Rewrite the expression.

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$