Precalculus Examples

$3^{a - 1} \cdot \frac{1}{81} = \frac{1}{9}$

Step 1

Combine $3^{a - 1}$ and $\frac{1}{81}$ .

$\frac{3^{a - 1}}{81} = \frac{1}{9}$

Step 2

Multiply both sides of the equation by $81$ .

$81 \frac{3^{a - 1}}{81} = 81 (\frac{1}{9})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $81$ .

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Step 3.1.1.1

Cancel the common factor.

$81 \frac{3^{a - 1}}{81} = 81 (\frac{1}{9})$

Step 3.1.1.2

Rewrite the expression.

$3^{a - 1} = 81 (\frac{1}{9})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Cancel the common factor of $9$ .

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Step 3.2.1.1

Factor $9$ out of $81$ .

$3^{a - 1} = 9 (9) \frac{1}{9}$

Step 3.2.1.2

Cancel the common factor.

$3^{a - 1} = 9 \cdot 9 \frac{1}{9}$

Step 3.2.1.3

Rewrite the expression.

$3^{a - 1} = 9$

Step 4

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

$3^{a - 1} = 3^{2}$

Step 5

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

$a - 1 = 2$

Step 6

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

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Step 6.1

Add $1$ to both sides of the equation.

$a = 2 + 1$

Step 6.2

Add $2$ and $1$ .

$a = 3$