Precalculus Examples

$36^{2 a} \cdot 216^{3 a} = {(\frac{1}{36})}^{- 2 a - 2}$

Step 1

Apply the product rule to $\frac{1}{36}$ .

$36^{2 a} \cdot 216^{3 a} = \frac{1^{- 2 a - 2}}{36^{- 2 a - 2}}$

Step 2

One to any power is one.

$36^{2 a} \cdot 216^{3 a} = \frac{1}{36^{- 2 a - 2}}$

Step 3

Move $36^{- 2 a - 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$36^{2 a} \cdot 216^{3 a} = 36^{- (- 2 a - 2)}$

Step 4

Rewrite $36$ as $6^{2}$ .

${(6^{2})}^{2 a} \cdot 216^{3 a} = 36^{- (- 2 a - 2)}$

Step 5

Multiply the exponents in ${(6^{2})}^{2 a}$ .

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Step 5.1

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

$6^{2 (2 a)} \cdot 216^{3 a} = 36^{- (- 2 a - 2)}$

Step 5.2

Multiply $2$ by $2$ .

$6^{4 a} \cdot 216^{3 a} = 36^{- (- 2 a - 2)}$

Step 6

Rewrite $216$ as $6^{3}$ .

$6^{4 a} \cdot {(6^{3})}^{3 a} = 36^{- (- 2 a - 2)}$

Step 7

Multiply the exponents in ${(6^{3})}^{3 a}$ .

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Step 7.1

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

$6^{4 a} \cdot 6^{3 (3 a)} = 36^{- (- 2 a - 2)}$

Step 7.2

Multiply $3$ by $3$ .

$6^{4 a} \cdot 6^{9 a} = 36^{- (- 2 a - 2)}$

Step 8

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

$6^{4 a + 9 a} = 36^{- (- 2 a - 2)}$

Step 9

Add $4 a$ and $9 a$ .

$6^{13 a} = 36^{- (- 2 a - 2)}$

Step 10

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

$6^{13 a} = 6^{2 (- (- 2 a - 2))}$

Step 11

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

$13 a = 2 (- (- 2 a - 2))$

Step 12

Solve for $a$ .

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Step 12.1

Simplify $2 (- (- 2 a - 2))$ .

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Step 12.1.1

Apply the distributive property.

$13 a = 2 (- (- 2 a) - - 2)$

Step 12.1.2

Multiply.

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Step 12.1.2.1

Multiply $- 2$ by $- 1$ .

$13 a = 2 (2 a - - 2)$

Step 12.1.2.2

Multiply $- 1$ by $- 2$ .

$13 a = 2 (2 a + 2)$

Step 12.1.3

Apply the distributive property.

$13 a = 2 (2 a) + 2 \cdot 2$

Step 12.1.4

Multiply.

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Step 12.1.4.1

Multiply $2$ by $2$ .

$13 a = 4 a + 2 \cdot 2$

Step 12.1.4.2

Multiply $2$ by $2$ .

$13 a = 4 a + 4$

Step 12.2

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

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Step 12.2.1

Subtract $4 a$ from both sides of the equation.

$13 a - 4 a = 4$

Step 12.2.2

Subtract $4 a$ from $13 a$ .

$9 a = 4$

Step 12.3

Divide each term in $9 a = 4$ by $9$ and simplify.

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Step 12.3.1

Divide each term in $9 a = 4$ by $9$ .

$\frac{9 a}{9} = \frac{4}{9}$

Step 12.3.2

Simplify the left side.

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Step 12.3.2.1

Cancel the common factor of $9$ .

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Step 12.3.2.1.1

Cancel the common factor.

$\frac{9 a}{9} = \frac{4}{9}$

Step 12.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{4}{9}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$a = \frac{4}{9}$

Decimal Form:

$a = 0.\overline{4}$