Precalculus Examples

$27^{- a} \cdot 81^{a} = 243^{3 a + 3}$

Step 1

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

${(3^{3})}^{- a} \cdot 81^{a} = 243^{3 a + 3}$

Step 2

Multiply the exponents in ${(3^{3})}^{- a}$ .

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Step 2.1

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

$3^{3 (- a)} \cdot 81^{a} = 243^{3 a + 3}$

Step 2.2

Multiply $- 1$ by $3$ .

$3^{- 3 a} \cdot 81^{a} = 243^{3 a + 3}$

Step 3

Rewrite $81$ as $3^{4}$ .

$3^{- 3 a} \cdot {(3^{4})}^{a} = 243^{3 a + 3}$

Step 4

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

$3^{- 3 a} \cdot 3^{4 a} = 243^{3 a + 3}$

Step 5

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

$3^{- 3 a + 4 a} = 243^{3 a + 3}$

Step 6

Add $- 3 a$ and $4 a$ .

$3^{a} = 243^{3 a + 3}$

Step 7

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

$3^{a} = 3^{5 (3 a + 3)}$

Step 8

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

$a = 5 (3 a + 3)$

Step 9

Solve for $a$ .

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Step 9.1

Simplify $5 (3 a + 3)$ .

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Step 9.1.1

Apply the distributive property.

$a = 5 (3 a) + 5 \cdot 3$

Step 9.1.2

Multiply.

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Step 9.1.2.1

Multiply $3$ by $5$ .

$a = 15 a + 5 \cdot 3$

Step 9.1.2.2

Multiply $5$ by $3$ .

$a = 15 a + 15$

Step 9.2

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

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Step 9.2.1

Subtract $15 a$ from both sides of the equation.

$a - 15 a = 15$

Step 9.2.2

Subtract $15 a$ from $a$ .

$- 14 a = 15$

Step 9.3

Divide each term in $- 14 a = 15$ by $- 14$ and simplify.

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Step 9.3.1

Divide each term in $- 14 a = 15$ by $- 14$ .

$\frac{- 14 a}{- 14} = \frac{15}{- 14}$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Cancel the common factor of $- 14$ .

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Step 9.3.2.1.1

Cancel the common factor.

$\frac{- 14 a}{- 14} = \frac{15}{- 14}$

Step 9.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{15}{- 14}$

Step 9.3.3

Simplify the right side.

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Step 9.3.3.1

Move the negative in front of the fraction.

$a = - \frac{15}{14}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{15}{14}$

Decimal Form:

$a = - 1.07142857 \dots$

Mixed Number Form:

$a = - 1 \frac{1}{14}$