Algebra Examples

${(\frac{1}{9})}^{a + 1} = 81^{a + 1} \cdot 27^{2 - a}$

Step 1

Subtract $81^{a + 1} \cdot 27^{2 - a}$ from both sides of the equation.

${(\frac{1}{9})}^{a + 1} - 81^{a + 1} \cdot 27^{2 - a} = 0$

Step 2

Simplify ${(\frac{1}{9})}^{a + 1} - 81^{a + 1} \cdot 27^{2 - a}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $\frac{1}{9}$ .

$\frac{1^{a + 1}}{9^{a + 1}} - 81^{a + 1} \cdot 27^{2 - a} = 0$

Step 2.1.2

One to any power is one.

$\frac{1}{9^{a + 1}} - 81^{a + 1} \cdot 27^{2 - a} = 0$

Step 2.1.3

Multiply $- 81^{a + 1} \cdot 27^{2 - a}$ .

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Step 2.1.3.1

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

$\frac{1}{9^{a + 1}} - ({(3^{3})}^{2 - a} \cdot 81^{a + 1}) = 0$

Step 2.1.3.2

Multiply the exponents in ${(3^{3})}^{2 - a}$ .

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Step 2.1.3.2.1

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

$\frac{1}{9^{a + 1}} - (3^{3 (2 - a)} \cdot 81^{a + 1}) = 0$

Step 2.1.3.2.2

Apply the distributive property.

$\frac{1}{9^{a + 1}} - (3^{3 \cdot 2 + 3 (- a)} \cdot 81^{a + 1}) = 0$

Step 2.1.3.2.3

Multiply $3$ by $2$ .

$\frac{1}{9^{a + 1}} - (3^{6 + 3 (- a)} \cdot 81^{a + 1}) = 0$

Step 2.1.3.2.4

Multiply $- 1$ by $3$ .

$\frac{1}{9^{a + 1}} - (3^{6 - 3 a} \cdot 81^{a + 1}) = 0$

Step 2.1.3.3

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

$\frac{1}{9^{a + 1}} - (3^{6 - 3 a} \cdot {(3^{4})}^{a + 1}) = 0$

Step 2.1.3.4

Multiply the exponents in ${(3^{4})}^{a + 1}$ .

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Step 2.1.3.4.1

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

$\frac{1}{9^{a + 1}} - (3^{6 - 3 a} \cdot 3^{4 (a + 1)}) = 0$

Step 2.1.3.4.2

Apply the distributive property.

$\frac{1}{9^{a + 1}} - (3^{6 - 3 a} \cdot 3^{4 a + 4 \cdot 1}) = 0$

Step 2.1.3.4.3

Multiply $4$ by $1$ .

$\frac{1}{9^{a + 1}} - (3^{6 - 3 a} \cdot 3^{4 a + 4}) = 0$

Step 2.1.3.5

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

$\frac{1}{9^{a + 1}} - 3^{6 - 3 a + 4 a + 4} = 0$

Step 2.1.3.6

Add $6$ and $4$ .

$\frac{1}{9^{a + 1}} - 3^{- 3 a + 4 a + 10} = 0$

Step 2.1.3.7

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

$\frac{1}{9^{a + 1}} - 3^{a + 10} = 0$

Step 2.2

To write $- 3^{a + 10}$ as a fraction with a common denominator, multiply by $\frac{9^{a + 1}}{9^{a + 1}}$ .

$\frac{1}{9^{a + 1}} - 3^{a + 10} \cdot \frac{9^{a + 1}}{9^{a + 1}} = 0$

Step 2.3

Combine $- 3^{a + 10}$ and $\frac{9^{a + 1}}{9^{a + 1}}$ .

$\frac{1}{9^{a + 1}} + \frac{- 3^{a + 10} \cdot 9^{a + 1}}{9^{a + 1}} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{1 - 3^{a + 10} \cdot 9^{a + 1}}{9^{a + 1}} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Rewrite $1$ as $1^{3}$ .

$\frac{1^{3} - 3^{a + 10} \cdot 9^{a + 1}}{9^{a + 1}} = 0$

Step 2.5.2

Rewrite $3^{a + 10} \cdot 9^{a + 1}$ as ${(3^{a + 4})}^{3}$ .

$\frac{1^{3} - {(3^{a + 4})}^{3}}{9^{a + 1}} = 0$

Step 2.5.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = 3^{a + 4}$ .

$\frac{(1 - 3^{a + 4}) (1^{2} + 1 \cdot 3^{a + 4} + {(3^{a + 4})}^{2})}{9^{a + 1}} = 0$

Step 2.5.4

Simplify.

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Step 2.5.4.1

One to any power is one.

$\frac{(1 - 3^{a + 4}) (1 + 1 \cdot 3^{a + 4} + {(3^{a + 4})}^{2})}{9^{a + 1}} = 0$

Step 2.5.4.2

Multiply $3^{a + 4}$ by $1$ .

$\frac{(1 - 3^{a + 4}) (1 + 3^{a + 4} + {(3^{a + 4})}^{2})}{9^{a + 1}} = 0$

Step 2.5.4.3

Multiply the exponents in ${(3^{a + 4})}^{2}$ .

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Step 2.5.4.3.1

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

$\frac{(1 - 3^{a + 4}) (1 + 3^{a + 4} + 3^{(a + 4) \cdot 2})}{9^{a + 1}} = 0$

Step 2.5.4.3.2

Apply the distributive property.

$\frac{(1 - 3^{a + 4}) (1 + 3^{a + 4} + 3^{a \cdot 2 + 4 \cdot 2})}{9^{a + 1}} = 0$

Step 2.5.4.3.3

Move $2$ to the left of $a$ .

$\frac{(1 - 3^{a + 4}) (1 + 3^{a + 4} + 3^{2 \cdot a + 4 \cdot 2})}{9^{a + 1}} = 0$

Step 2.5.4.3.4

Multiply $4$ by $2$ .

$\frac{(1 - 3^{a + 4}) (1 + 3^{a + 4} + 3^{2 a + 8})}{9^{a + 1}} = 0$

Step 3

Set the numerator equal to zero.

$(1 - 3^{a + 4}) (1 + 3^{a + 4} + 3^{2 a + 8}) = 0$

Step 4

Solve the equation for $a$ .

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Step 4.1

Simplify $(1 - 3^{a + 4}) (1 + 3^{a + 4} + 3^{2 a + 8})$ .

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Step 4.1.1

Expand $(1 - 3^{a + 4}) (1 + 3^{a + 4} + 3^{2 a + 8})$ by multiplying each term in the first expression by each term in the second expression.

$1 \cdot 1 + 1 \cdot 3^{a + 4} + 1 \cdot 3^{2 a + 8} - 3^{a + 4} \cdot 1 - 3^{a + 4} \cdot 3^{a + 4} - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2

Simplify terms.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Multiply $1$ by $1$ .

$1 + 1 \cdot 3^{a + 4} + 1 \cdot 3^{2 a + 8} - 3^{a + 4} \cdot 1 - 3^{a + 4} \cdot 3^{a + 4} - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2.1.2

Multiply $3^{a + 4}$ by $1$ .

$1 + 3^{a + 4} + 1 \cdot 3^{2 a + 8} - 3^{a + 4} \cdot 1 - 3^{a + 4} \cdot 3^{a + 4} - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2.1.3

Multiply $3^{2 a + 8}$ by $1$ .

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} \cdot 1 - 3^{a + 4} \cdot 3^{a + 4} - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2.1.4

Multiply $- 1$ by $1$ .

$1 + 3^{a + 4} + 3^{2 a + 8} - 1 \cdot 3^{a + 4} - 3^{a + 4} \cdot 3^{a + 4} - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2.1.5

Rewrite $- 1 \cdot 3^{a + 4}$ as $- 3^{a + 4}$ .

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - 3^{a + 4} \cdot 3^{a + 4} - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2.1.6

Multiply $3^{a + 4}$ by $3^{a + 4}$ by adding the exponents.

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Step 4.1.2.1.6.1

Move $3^{a + 4}$ .

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - (3^{a + 4} \cdot 3^{a + 4}) - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2.1.6.2

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

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - 3^{a + 4 + a + 4} - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2.1.6.3

Add $a$ and $a$ .

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - 3^{2 a + 4 + 4} - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2.1.6.4

Add $4$ and $4$ .

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - 3^{2 a + 8} - 3^{a + 4} \cdot 3^{2 a + 8} = 0$

Step 4.1.2.1.7

Multiply $3^{a + 4}$ by $3^{2 a + 8}$ by adding the exponents.

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Step 4.1.2.1.7.1

Move $3^{2 a + 8}$ .

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - 3^{2 a + 8} - (3^{2 a + 8} \cdot 3^{a + 4}) = 0$

Step 4.1.2.1.7.2

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

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - 3^{2 a + 8} - 3^{2 a + 8 + a + 4} = 0$

Step 4.1.2.1.7.3

Add $2 a$ and $a$ .

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - 3^{2 a + 8} - 3^{3 a + 8 + 4} = 0$

Step 4.1.2.1.7.4

Add $8$ and $4$ .

$1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - 3^{2 a + 8} - 3^{3 a + 12} = 0$

Step 4.1.2.2

Combine the opposite terms in $1 + 3^{a + 4} + 3^{2 a + 8} - 3^{a + 4} - 3^{2 a + 8} - 3^{3 a + 12}$ .

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Step 4.1.2.2.1

Subtract $3^{a + 4}$ from $3^{a + 4}$ .

$1 + 0 + 3^{2 a + 8} - 3^{2 a + 8} - 3^{3 a + 12} = 0$

Step 4.1.2.2.2

Add $1$ and $0$ .

$1 + 3^{2 a + 8} - 3^{2 a + 8} - 3^{3 a + 12} = 0$

Step 4.1.2.2.3

Subtract $3^{2 a + 8}$ from $3^{2 a + 8}$ .

$1 + 0 - 3^{3 a + 12} = 0$

Step 4.1.2.2.4

Add $1$ and $0$ .

$1 - 3^{3 a + 12} = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$- 3^{3 a + 12} = - 1$

Step 4.3

Divide each term in $- 3^{3 a + 12} = - 1$ by $- 1$ and simplify.

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Step 4.3.1

Divide each term in $- 3^{3 a + 12} = - 1$ by $- 1$ .

$\frac{- 3^{3 a + 12}}{- 1} = \frac{- 1}{- 1}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{3^{3 a + 12}}{1} = \frac{- 1}{- 1}$

Step 4.3.2.2

Divide $3^{3 a + 12}$ by $1$ .

$3^{3 a + 12} = \frac{- 1}{- 1}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $- 1$ by $- 1$ .

$3^{3 a + 12} = 1$

Step 4.4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{3 a + 12}) = \ln (1)$

Step 4.5

Expand $\ln (3^{3 a + 12})$ by moving $3 a + 12$ outside the logarithm.

$(3 a + 12) \ln (3) = \ln (1)$

Step 4.6

Simplify the left side.

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Step 4.6.1

Apply the distributive property.

$3 a \ln (3) + 12 \ln (3) = \ln (1)$

Step 4.7

Simplify the right side.

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Step 4.7.1

The natural logarithm of $1$ is $0$ .

$3 a \ln (3) + 12 \ln (3) = 0$

Step 4.8

Subtract $12 \ln (3)$ from both sides of the equation.

$3 a \ln (3) = - 12 \ln (3)$

Step 4.9

Divide each term in $3 a \ln (3) = - 12 \ln (3)$ by $3 \ln (3)$ and simplify.

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Step 4.9.1

Divide each term in $3 a \ln (3) = - 12 \ln (3)$ by $3 \ln (3)$ .

$\frac{3 a \ln (3)}{3 \ln (3)} = \frac{- 12 \ln (3)}{3 \ln (3)}$

Step 4.9.2

Simplify the left side.

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Step 4.9.2.1

Cancel the common factor of $3$ .

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Step 4.9.2.1.1

Cancel the common factor.

$\frac{3 a \ln (3)}{3 \ln (3)} = \frac{- 12 \ln (3)}{3 \ln (3)}$

Step 4.9.2.1.2

Rewrite the expression.

$\frac{a \ln (3)}{\ln (3)} = \frac{- 12 \ln (3)}{3 \ln (3)}$

Step 4.9.2.2

Cancel the common factor of $\ln (3)$ .

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Step 4.9.2.2.1

Cancel the common factor.

$\frac{a \ln (3)}{\ln (3)} = \frac{- 12 \ln (3)}{3 \ln (3)}$

Step 4.9.2.2.2

Divide $a$ by $1$ .

$a = \frac{- 12 \ln (3)}{3 \ln (3)}$

Step 4.9.3

Simplify the right side.

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Step 4.9.3.1

Cancel the common factor of $- 12$ and $3$ .

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Step 4.9.3.1.1

Factor $3$ out of $- 12 \ln (3)$ .

$a = \frac{3 (- 4 \ln (3))}{3 \ln (3)}$

Step 4.9.3.1.2

Cancel the common factors.

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Step 4.9.3.1.2.1

Factor $3$ out of $3 \ln (3)$ .

$a = \frac{3 (- 4 \ln (3))}{3 \ln (3)}$

Step 4.9.3.1.2.2

Cancel the common factor.

$a = \frac{3 (- 4 \ln (3))}{3 \ln (3)}$

Step 4.9.3.1.2.3

Rewrite the expression.

$a = \frac{- 4 \ln (3)}{\ln (3)}$

Step 4.9.3.2

Cancel the common factor of $\ln (3)$ .

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Step 4.9.3.2.1

Cancel the common factor.

$a = \frac{- 4 \ln (3)}{\ln (3)}$

Step 4.9.3.2.2

Divide $- 4$ by $1$ .

$a = - 4$