Algebra Examples

$\frac{9^{3 n + 3}}{\frac{1}{27}} = 9$

Step 1

Raise $\frac{1}{27}$ to the power of $1$ .

$\frac{9^{3 n + 3}}{{(\frac{1}{27})}^{1}} = 9$

Step 2

Move ${(\frac{1}{27})}^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$9^{3 n + 3} {(\frac{1}{27})}^{- 1} = 9$

Step 3

Rewrite $9$ as $3^{2}$ .

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

Step 4

Multiply the exponents in ${(3^{2})}^{3 n + 3}$ .

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Step 4.1

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

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Multiply $3$ by $2$ .

$3^{6 n + 2 \cdot 3} {(\frac{1}{27})}^{- 1} = 9$

Step 4.4

Multiply $2$ by $3$ .

$3^{6 n + 6} {(\frac{1}{27})}^{- 1} = 9$

Step 5

Rewrite $\frac{1}{27}$ as $27^{- 1}$ .

$3^{6 n + 6} {(27^{- 1})}^{- 1} = 9$

Step 6

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

$3^{6 n + 6} {({(3^{3})}^{- 1})}^{- 1} = 9$

Step 7

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

$3^{6 n + 6} {(3^{3 \cdot - 1})}^{- 1} = 9$

Step 8

Multiply $3$ by $- 1$ .

$3^{6 n + 6} {(3^{- 3})}^{- 1} = 9$

Step 9

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

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Step 9.1

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

$3^{6 n + 6} \cdot 3^{- 3 \cdot - 1} = 9$

Step 9.2

Multiply $- 3$ by $- 1$ .

$3^{6 n + 6} \cdot 3^{3} = 9$

Step 10

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

$3^{6 n + 6 + 3} = 9$

Step 11

Add $6$ and $3$ .

$3^{6 n + 9} = 9$

Step 12

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

$3^{6 n + 9} = 3^{2}$

Step 13

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

$6 n + 9 = 2$

Step 14

Solve for $n$ .

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Step 14.1

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

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Step 14.1.1

Subtract $9$ from both sides of the equation.

$6 n = 2 - 9$

Step 14.1.2

Subtract $9$ from $2$ .

$6 n = - 7$

Step 14.2

Divide each term in $6 n = - 7$ by $6$ and simplify.

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Step 14.2.1

Divide each term in $6 n = - 7$ by $6$ .

$\frac{6 n}{6} = \frac{- 7}{6}$

Step 14.2.2

Simplify the left side.

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Step 14.2.2.1

Cancel the common factor of $6$ .

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Step 14.2.2.1.1

Cancel the common factor.

$\frac{6 n}{6} = \frac{- 7}{6}$

Step 14.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 7}{6}$

Step 14.2.3

Simplify the right side.

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Step 14.2.3.1

Move the negative in front of the fraction.

$n = - \frac{7}{6}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{7}{6}$

Decimal Form:

$n = - 1.1\overline{6}$

Mixed Number Form:

$n = - 1 \frac{1}{6}$