Algebra Examples

${(\frac{a^{3} b^{4}}{a^{2} b})}^{6} = a^{x} b^{y}$

Step 1

Rewrite the equation as $a^{x} b^{y} = {(\frac{a^{3} b^{4}}{a^{2} b})}^{6}$ .

$a^{x} b^{y} = {(\frac{a^{3} b^{4}}{a^{2} b})}^{6}$

Step 2

Simplify.

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Step 2.1

Reduce the expression $\frac{a^{3} b^{4}}{a^{2} b}$ by cancelling the common factors.

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Step 2.1.1

Factor $a^{2}$ out of $a^{3} b^{4}$ .

$a^{x} b^{y} = {(\frac{a^{2} (a b^{4})}{a^{2} b})}^{6}$

Step 2.1.2

Factor $a^{2}$ out of $a^{2} b$ .

$a^{x} b^{y} = {(\frac{a^{2} (a b^{4})}{a^{2} (b)})}^{6}$

Step 2.1.3

Cancel the common factor.

$a^{x} b^{y} = {(\frac{a^{2} (a b^{4})}{a^{2} b})}^{6}$

Step 2.1.4

Rewrite the expression.

$a^{x} b^{y} = {(\frac{a b^{4}}{b})}^{6}$

Step 2.2

Reduce the expression $\frac{a b^{4}}{b}$ by cancelling the common factors.

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Step 2.2.1

Factor $b$ out of $a b^{4}$ .

$a^{x} b^{y} = {(\frac{b (a b^{3})}{b})}^{6}$

Step 2.2.2

Raise $b$ to the power of $1$ .

$a^{x} b^{y} = {(\frac{b (a b^{3})}{b^{1}})}^{6}$

Step 2.2.3

Factor $b$ out of $b^{1}$ .

$a^{x} b^{y} = {(\frac{b (a b^{3})}{b \cdot 1})}^{6}$

Step 2.2.4

Cancel the common factor.

$a^{x} b^{y} = {(\frac{b (a b^{3})}{b \cdot 1})}^{6}$

Step 2.2.5

Rewrite the expression.

$a^{x} b^{y} = {(\frac{a b^{3}}{1})}^{6}$

Step 2.3

Divide $a b^{3}$ by $1$ .

$a^{x} b^{y} = {(a b^{3})}^{6}$

Step 3

Divide each term in $a^{x} b^{y} = {(a b^{3})}^{6}$ by $b^{y}$ and simplify.

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Step 3.1

Divide each term in $a^{x} b^{y} = {(a b^{3})}^{6}$ by $b^{y}$ .

$\frac{a^{x} b^{y}}{b^{y}} = \frac{{(a b^{3})}^{6}}{b^{y}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $b^{y}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{a^{x} b^{y}}{b^{y}} = \frac{{(a b^{3})}^{6}}{b^{y}}$

Step 3.2.1.2

Divide $a^{x}$ by $1$ .

$a^{x} = \frac{{(a b^{3})}^{6}}{b^{y}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Apply the product rule to $a b^{3}$ .

$a^{x} = \frac{a^{6} {(b^{3})}^{6}}{b^{y}}$

Step 3.3.1.2

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

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Step 3.3.1.2.1

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

$a^{x} = \frac{a^{6} b^{3 \cdot 6}}{b^{y}}$

Step 3.3.1.2.2

Multiply $3$ by $6$ .

$a^{x} = \frac{a^{6} b^{18}}{b^{y}}$

Step 3.3.2

Cancel the common factor of $b^{18}$ and $b^{y}$ .

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Step 3.3.2.1

Factor $b^{y}$ out of $a^{6} b^{18}$ .

$a^{x} = \frac{b^{y} (a^{6} b^{18 - y})}{b^{y}}$

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

Multiply by $1$ .

$a^{x} = \frac{b^{y} (a^{6} b^{18 - y})}{b^{y} \cdot 1}$

Step 3.3.2.2.2

Cancel the common factor.

$a^{x} = \frac{b^{y} (a^{6} b^{18 - y})}{b^{y} \cdot 1}$

Step 3.3.2.2.3

Rewrite the expression.

$a^{x} = \frac{a^{6} b^{18 - y}}{1}$

Step 3.3.2.2.4

Divide $a^{6} b^{18 - y}$ by $1$ .

$a^{x} = a^{6} b^{18 - y}$

Step 4

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

$\ln (a^{x}) = \ln (a^{6} b^{18 - y})$

Step 5

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

$x \ln (a) = \ln (a^{6} b^{18 - y})$

Step 6

Expand the right side.

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Step 6.1

Rewrite $\ln (a^{6} b^{18 - y})$ as $\ln (a^{6}) + \ln (b^{18 - y})$ .

$x \ln (a) = \ln (a^{6}) + \ln (b^{18 - y})$

Step 6.2

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

$x \ln (a) = 6 \ln (a) + \ln (b^{18 - y})$

Step 6.3

Expand $\ln (b^{18 - y})$ by moving $18 - y$ outside the logarithm.

$x \ln (a) = 6 \ln (a) + (18 - y) \ln (b)$

Step 7

Simplify the right side.

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Step 7.1

Apply the distributive property.

$x \ln (a) = 6 \ln (a) + 18 \ln (b) - y \ln (b)$

Step 8

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

$x \ln (a) - 6 \ln (a) - 18 \ln (b) + y \ln (b) = 0$

Step 9

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

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Step 9.1

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

$x \ln (a) - 18 \ln (b) + y \ln (b) = 6 \ln (a)$

Step 9.2

Add $18 \ln (b)$ to both sides of the equation.

$x \ln (a) + y \ln (b) = 6 \ln (a) + 18 \ln (b)$

Step 9.3

Subtract $y \ln (b)$ from both sides of the equation.

$x \ln (a) = 6 \ln (a) + 18 \ln (b) - y \ln (b)$

Step 10

Divide each term in $x \ln (a) = 6 \ln (a) + 18 \ln (b) - y \ln (b)$ by $\ln (a)$ and simplify.

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Step 10.1

Divide each term in $x \ln (a) = 6 \ln (a) + 18 \ln (b) - y \ln (b)$ by $\ln (a)$ .

$\frac{x \ln (a)}{\ln (a)} = \frac{6 \ln (a)}{\ln (a)} + \frac{18 \ln (b)}{\ln (a)} + \frac{- y \ln (b)}{\ln (a)}$

Step 10.2

Simplify the left side.

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Step 10.2.1

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

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Step 10.2.1.1

Cancel the common factor.

$\frac{x \ln (a)}{\ln (a)} = \frac{6 \ln (a)}{\ln (a)} + \frac{18 \ln (b)}{\ln (a)} + \frac{- y \ln (b)}{\ln (a)}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{6 \ln (a)}{\ln (a)} + \frac{18 \ln (b)}{\ln (a)} + \frac{- y \ln (b)}{\ln (a)}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Simplify each term.

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Step 10.3.1.1

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

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Step 10.3.1.1.1

Cancel the common factor.

$x = \frac{6 \ln (a)}{\ln (a)} + \frac{18 \ln (b)}{\ln (a)} + \frac{- y \ln (b)}{\ln (a)}$

Step 10.3.1.1.2

Divide $6$ by $1$ .

$x = 6 + \frac{18 \ln (b)}{\ln (a)} + \frac{- y \ln (b)}{\ln (a)}$

Step 10.3.1.2

Move the negative in front of the fraction.

$x = 6 + \frac{18 \ln (b)}{\ln (a)} - \frac{y \ln (b)}{\ln (a)}$