Precalculus Examples

$81^{3 k} \cdot 81^{1 - 2 k} = 9$

Step 1

Multiply $81^{3 k}$ by $81^{1 - 2 k}$ 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.

$81^{3 k + 1 - 2 k} = 9$

Step 1.2

Subtract $2 k$ from $3 k$ .

$81^{k + 1} = 9$

Step 2

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

$9^{2 (k + 1)} = 9^{1}$

Step 3

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

$2 (k + 1) = 1$

Step 4

Solve for $k$ .

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Step 4.1

Divide each term in $2 (k + 1) = 1$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (k + 1) = 1$ by $2$ .

$\frac{2 (k + 1)}{2} = \frac{1}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (k + 1)}{2} = \frac{1}{2}$

Step 4.1.2.1.2

Divide $k + 1$ by $1$ .

$k + 1 = \frac{1}{2}$

Step 4.2

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

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Step 4.2.1

Subtract $1$ from both sides of the equation.

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

Step 4.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$k = \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 4.2.3

Combine $- 1$ and $\frac{2}{2}$ .

$k = \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 4.2.4

Combine the numerators over the common denominator.

$k = \frac{1 - 1 \cdot 2}{2}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 1$ by $2$ .

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

Step 4.2.5.2

Subtract $2$ from $1$ .

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

Step 4.2.6

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$k = - 0.5$