Algebra Examples

$\sqrt[n - 1]{27^{5} {(x^{9} y^{3})}^{2}} = 243 x^{6} y^{2}$

Step 1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $n - 1$ .

${\sqrt[n - 1]{27^{5} {(x^{9} y^{3})}^{2}}}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[n - 1]{27^{5} {(x^{9} y^{3})}^{2}}$ as ${(27^{5} {(x^{9} y^{3})}^{2})}^{\frac{1}{n - 1}}$ .

${({(27^{5} {(x^{9} y^{3})}^{2})}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(27^{5} {(x^{9} y^{3})}^{2})}^{\frac{1}{n - 1}})}^{n - 1}$ .

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Step 2.2.1.1

Raise $27$ to the power of $5$ .

${({(14348907 {(x^{9} y^{3})}^{2})}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.2

Apply the product rule to $x^{9} y^{3}$ .

${({(14348907 ({(x^{9})}^{2} {(y^{3})}^{2}))}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.3

Multiply the exponents in ${(x^{9})}^{2}$ .

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Step 2.2.1.3.1

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

${({(14348907 (x^{9 \cdot 2} {(y^{3})}^{2}))}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.3.2

Multiply $9$ by $2$ .

${({(14348907 (x^{18} {(y^{3})}^{2}))}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.4

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

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Step 2.2.1.4.1

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

${({(14348907 (x^{18} y^{3 \cdot 2}))}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.4.2

Multiply $3$ by $2$ .

${({(14348907 (x^{18} y^{6}))}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.1.5.1

Apply the product rule to $14348907 x^{18} y^{6}$ .

${({(14348907 x^{18})}^{\frac{1}{n - 1}} {(y^{6})}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.5.2

Apply the product rule to $14348907 x^{18}$ .

${(14348907^{\frac{1}{n - 1}} {(x^{18})}^{\frac{1}{n - 1}} {(y^{6})}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.6

Multiply the exponents in ${(x^{18})}^{\frac{1}{n - 1}}$ .

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Step 2.2.1.6.1

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

${(14348907^{\frac{1}{n - 1}} x^{18 \frac{1}{n - 1}} {(y^{6})}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.6.2

Combine $18$ and $\frac{1}{n - 1}$ .

${(14348907^{\frac{1}{n - 1}} x^{\frac{18}{n - 1}} {(y^{6})}^{\frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.7

Multiply the exponents in ${(y^{6})}^{\frac{1}{n - 1}}$ .

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Step 2.2.1.7.1

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

${(14348907^{\frac{1}{n - 1}} x^{\frac{18}{n - 1}} y^{6 \frac{1}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.7.2

Combine $6$ and $\frac{1}{n - 1}$ .

${(14348907^{\frac{1}{n - 1}} x^{\frac{18}{n - 1}} y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.1.8.1

Apply the product rule to $14348907^{\frac{1}{n - 1}} x^{\frac{18}{n - 1}} y^{\frac{6}{n - 1}}$ .

${(14348907^{\frac{1}{n - 1}} x^{\frac{18}{n - 1}})}^{n - 1} {(y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.8.2

Apply the product rule to $14348907^{\frac{1}{n - 1}} x^{\frac{18}{n - 1}}$ .

${(14348907^{\frac{1}{n - 1}})}^{n - 1} {(x^{\frac{18}{n - 1}})}^{n - 1} {(y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.9

Multiply the exponents in ${(14348907^{\frac{1}{n - 1}})}^{n - 1}$ .

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Step 2.2.1.9.1

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

$14348907^{\frac{1}{n - 1} (n - 1)} {(x^{\frac{18}{n - 1}})}^{n - 1} {(y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.9.2

Cancel the common factor of $n - 1$ .

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Step 2.2.1.9.2.1

Cancel the common factor.

$14348907^{\frac{1}{n - 1} (n - 1)} {(x^{\frac{18}{n - 1}})}^{n - 1} {(y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.9.2.2

Rewrite the expression.

$14348907^{1} {(x^{\frac{18}{n - 1}})}^{n - 1} {(y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.10

Evaluate the exponent.

$14348907 {(x^{\frac{18}{n - 1}})}^{n - 1} {(y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.11

Multiply the exponents in ${(x^{\frac{18}{n - 1}})}^{n - 1}$ .

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Step 2.2.1.11.1

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

$14348907 x^{\frac{18}{n - 1} (n - 1)} {(y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.11.2

Cancel the common factor of $n - 1$ .

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Step 2.2.1.11.2.1

Cancel the common factor.

$14348907 x^{\frac{18}{n - 1} (n - 1)} {(y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.11.2.2

Rewrite the expression.

$14348907 x^{18} {(y^{\frac{6}{n - 1}})}^{n - 1} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.12

Multiply the exponents in ${(y^{\frac{6}{n - 1}})}^{n - 1}$ .

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Step 2.2.1.12.1

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

$14348907 x^{18} y^{\frac{6}{n - 1} (n - 1)} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.12.2

Cancel the common factor of $n - 1$ .

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Step 2.2.1.12.2.1

Cancel the common factor.

$14348907 x^{18} y^{\frac{6}{n - 1} (n - 1)} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.2.1.12.2.2

Rewrite the expression.

$14348907 x^{18} y^{6} = {(243 x^{6} y^{2})}^{n - 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(243 x^{6} y^{2})}^{n - 1}$ .

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Step 2.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.3.1.1.1

Apply the product rule to $243 x^{6} y^{2}$ .

$14348907 x^{18} y^{6} = {(243 x^{6})}^{n - 1} {(y^{2})}^{n - 1}$

Step 2.3.1.1.2

Apply the product rule to $243 x^{6}$ .

$14348907 x^{18} y^{6} = 243^{n - 1} {(x^{6})}^{n - 1} {(y^{2})}^{n - 1}$

Step 2.3.1.2

Multiply the exponents in ${(x^{6})}^{n - 1}$ .

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Step 2.3.1.2.1

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

$14348907 x^{18} y^{6} = 243^{n - 1} x^{6 (n - 1)} {(y^{2})}^{n - 1}$

Step 2.3.1.2.2

Apply the distributive property.

$14348907 x^{18} y^{6} = 243^{n - 1} x^{6 n + 6 \cdot - 1} {(y^{2})}^{n - 1}$

Step 2.3.1.2.3

Multiply $6$ by $- 1$ .

$14348907 x^{18} y^{6} = 243^{n - 1} x^{6 n - 6} {(y^{2})}^{n - 1}$

Step 2.3.1.3

Multiply the exponents in ${(y^{2})}^{n - 1}$ .

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Step 2.3.1.3.1

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

$14348907 x^{18} y^{6} = 243^{n - 1} x^{6 n - 6} y^{2 (n - 1)}$

Step 2.3.1.3.2

Apply the distributive property.

$14348907 x^{18} y^{6} = 243^{n - 1} x^{6 n - 6} y^{2 n + 2 \cdot - 1}$

Step 2.3.1.3.3

Multiply $2$ by $- 1$ .

$14348907 x^{18} y^{6} = 243^{n - 1} x^{6 n - 6} y^{2 n - 2}$

Step 3

Solve for $n$ .

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Step 3.1

Rewrite the equation as $243^{n - 1} x^{6 n - 6} y^{2 n - 2} = 14348907 x^{18} y^{6}$ .

$243^{n - 1} x^{6 n - 6} y^{2 n - 2} = 14348907 x^{18} y^{6}$

Step 3.2

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

$\ln (243^{n - 1} x^{6 n - 6} y^{2 n - 2}) = \ln (14348907 x^{18} y^{6})$

Step 3.3

Expand the left side.

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Step 3.3.1

Rewrite $\ln (243^{n - 1} x^{6 n - 6} y^{2 n - 2})$ as $\ln (243^{n - 1} x^{6 n - 6}) + \ln (y^{2 n - 2})$ .

$\ln (243^{n - 1} x^{6 n - 6}) + \ln (y^{2 n - 2}) = \ln (14348907 x^{18} y^{6})$

Step 3.3.2

Rewrite $\ln (243^{n - 1} x^{6 n - 6})$ as $\ln (243^{n - 1}) + \ln (x^{6 n - 6})$ .

$\ln (243^{n - 1}) + \ln (x^{6 n - 6}) + \ln (y^{2 n - 2}) = \ln (14348907 x^{18} y^{6})$

Step 3.3.3

Expand $\ln (243^{n - 1})$ by moving $n - 1$ outside the logarithm.

$(n - 1) \ln (243) + \ln (x^{6 n - 6}) + \ln (y^{2 n - 2}) = \ln (14348907 x^{18} y^{6})$

Step 3.3.4

Expand $\ln (x^{6 n - 6})$ by moving $6 n - 6$ outside the logarithm.

$(n - 1) \ln (243) + (6 n - 6) \ln (x) + \ln (y^{2 n - 2}) = \ln (14348907 x^{18} y^{6})$

Step 3.3.5

Expand $\ln (y^{2 n - 2})$ by moving $2 n - 2$ outside the logarithm.

$(n - 1) \ln (243) + (6 n - 6) \ln (x) + (2 n - 2) \ln (y) = \ln (14348907 x^{18} y^{6})$

Step 3.4

Simplify the left side.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Apply the distributive property.

$n \ln (243) - 1 \ln (243) + (6 n - 6) \ln (x) + (2 n - 2) \ln (y) = \ln (14348907 x^{18} y^{6})$

Step 3.4.1.2

Rewrite $- 1 \ln (243)$ as $- \ln (243)$ .

$n \ln (243) - \ln (243) + (6 n - 6) \ln (x) + (2 n - 2) \ln (y) = \ln (14348907 x^{18} y^{6})$

Step 3.4.1.3

Apply the distributive property.

$n \ln (243) - \ln (243) + 6 n \ln (x) - 6 \ln (x) + (2 n - 2) \ln (y) = \ln (14348907 x^{18} y^{6})$

Step 3.4.1.4

Apply the distributive property.

$n \ln (243) - \ln (243) + 6 n \ln (x) - 6 \ln (x) + 2 n \ln (y) - 2 \ln (y) = \ln (14348907 x^{18} y^{6})$

Step 3.5

Simplify the left side.

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Step 3.5.1

Move $- 6 \ln (x)$ .

$n \ln (243) - \ln (243) + 6 n \ln (x) + 2 n \ln (y) - 6 \ln (x) - 2 \ln (y) = \ln (14348907 x^{18} y^{6})$

Step 3.5.2

Move $- \ln (243)$ .

$n \ln (243) + 6 n \ln (x) + 2 n \ln (y) - \ln (243) - 6 \ln (x) - 2 \ln (y) = \ln (14348907 x^{18} y^{6})$

Step 3.6

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

$n \ln (243) + 6 n \ln (x) + 2 n \ln (y) - \ln (243) - 6 \ln (x) - 2 \ln (y) - \ln (14348907 x^{18} y^{6}) = 0$

Step 3.7

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

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Step 3.7.1

Add $\ln (243)$ to both sides of the equation.

$n \ln (243) + 6 n \ln (x) + 2 n \ln (y) - 6 \ln (x) - 2 \ln (y) - \ln (14348907 x^{18} y^{6}) = \ln (243)$

Step 3.7.2

Add $6 \ln (x)$ to both sides of the equation.

$n \ln (243) + 6 n \ln (x) + 2 n \ln (y) - 2 \ln (y) - \ln (14348907 x^{18} y^{6}) = \ln (243) + 6 \ln (x)$

Step 3.7.3

Add $2 \ln (y)$ to both sides of the equation.

$n \ln (243) + 6 n \ln (x) + 2 n \ln (y) - \ln (14348907 x^{18} y^{6}) = \ln (243) + 6 \ln (x) + 2 \ln (y)$

Step 3.7.4

Add $\ln (14348907 x^{18} y^{6})$ to both sides of the equation.

$n \ln (243) + 6 n \ln (x) + 2 n \ln (y) = \ln (243) + 6 \ln (x) + 2 \ln (y) + \ln (14348907 x^{18} y^{6})$

Step 3.8

Factor $n$ out of $n \ln (243) + 6 n \ln (x) + 2 n \ln (y)$ .

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Step 3.8.1

Factor $n$ out of $n \ln (243)$ .

$n (\ln (243)) + 6 n \ln (x) + 2 n \ln (y) = \ln (243) + 6 \ln (x) + 2 \ln (y) + \ln (14348907 x^{18} y^{6})$

Step 3.8.2

Factor $n$ out of $6 n \ln (x)$ .

$n (\ln (243)) + n (6 \ln (x)) + 2 n \ln (y) = \ln (243) + 6 \ln (x) + 2 \ln (y) + \ln (14348907 x^{18} y^{6})$

Step 3.8.3

Factor $n$ out of $2 n \ln (y)$ .

$n (\ln (243)) + n (6 \ln (x)) + n (2 \ln (y)) = \ln (243) + 6 \ln (x) + 2 \ln (y) + \ln (14348907 x^{18} y^{6})$

Step 3.8.4

Factor $n$ out of $n (\ln (243)) + n (6 \ln (x))$ .

$n (\ln (243) + 6 \ln (x)) + n (2 \ln (y)) = \ln (243) + 6 \ln (x) + 2 \ln (y) + \ln (14348907 x^{18} y^{6})$

Step 3.8.5

Factor $n$ out of $n (\ln (243) + 6 \ln (x)) + n (2 \ln (y))$ .

$n (\ln (243) + 6 \ln (x) + 2 \ln (y)) = \ln (243) + 6 \ln (x) + 2 \ln (y) + \ln (14348907 x^{18} y^{6})$

Step 3.9

Divide each term in $n (\ln (243) + 6 \ln (x) + 2 \ln (y)) = \ln (243) + 6 \ln (x) + 2 \ln (y) + \ln (14348907 x^{18} y^{6})$ by $\ln (243) + 6 \ln (x) + 2 \ln (y)$ and simplify.

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Step 3.9.1

Divide each term in $n (\ln (243) + 6 \ln (x) + 2 \ln (y)) = \ln (243) + 6 \ln (x) + 2 \ln (y) + \ln (14348907 x^{18} y^{6})$ by $\ln (243) + 6 \ln (x) + 2 \ln (y)$ .

$\frac{n (\ln (243) + 6 \ln (x) + 2 \ln (y))}{\ln (243) + 6 \ln (x) + 2 \ln (y)} = \frac{\ln (243)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{6 \ln (x)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{2 \ln (y)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{\ln (14348907 x^{18} y^{6})}{\ln (243) + 6 \ln (x) + 2 \ln (y)}$

Step 3.9.2

Simplify the left side.

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Step 3.9.2.1

Cancel the common factor of $\ln (243) + 6 \ln (x) + 2 \ln (y)$ .

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Step 3.9.2.1.1

Cancel the common factor.

Step 3.9.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (243)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{6 \ln (x)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{2 \ln (y)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{\ln (14348907 x^{18} y^{6})}{\ln (243) + 6 \ln (x) + 2 \ln (y)}$

Step 3.9.3

Simplify the right side.

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Step 3.9.3.1

Combine the numerators over the common denominator.

$n = \frac{\ln (243) + 6 \ln (x)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{2 \ln (y)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{\ln (14348907 x^{18} y^{6})}{\ln (243) + 6 \ln (x) + 2 \ln (y)}$

Step 3.9.3.2

Combine the numerators over the common denominator.

$n = \frac{\ln (243) + 6 \ln (x) + 2 \ln (y)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{\ln (14348907 x^{18} y^{6})}{\ln (243) + 6 \ln (x) + 2 \ln (y)}$

Step 3.9.3.3

Cancel the common factor of $\ln (243) + 6 \ln (x) + 2 \ln (y)$ .

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Step 3.9.3.3.1

Cancel the common factor.

$n = \frac{\ln (243) + 6 \ln (x) + 2 \ln (y)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{\ln (14348907 x^{18} y^{6})}{\ln (243) + 6 \ln (x) + 2 \ln (y)}$

Step 3.9.3.3.2

Rewrite the expression.

$n = 1 + \frac{\ln (14348907 x^{18} y^{6})}{\ln (243) + 6 \ln (x) + 2 \ln (y)}$

Step 3.9.3.4

Write $1$ as a fraction with a common denominator.

$n = \frac{\ln (243) + 6 \ln (x) + 2 \ln (y)}{\ln (243) + 6 \ln (x) + 2 \ln (y)} + \frac{\ln (14348907 x^{18} y^{6})}{\ln (243) + 6 \ln (x) + 2 \ln (y)}$

Step 3.9.3.5

Combine the numerators over the common denominator.

$n = \frac{\ln (243) + 6 \ln (x) + 2 \ln (y) + \ln (14348907 x^{18} y^{6})}{\ln (243) + 6 \ln (x) + 2 \ln (y)}$