Algebra Examples

$\frac{x^{2} {(y^{3})}^{4}}{x y^{5}} = x^{a} \cdot y^{b}$

Step 1

Rewrite the equation as $x^{a} \cdot y^{b} = \frac{x^{2} {(y^{3})}^{4}}{x y^{5}}$ .

$x^{a} \cdot y^{b} = \frac{x^{2} {(y^{3})}^{4}}{x y^{5}}$

Step 2

Reduce the expression $\frac{x^{2} {(y^{3})}^{4}}{x y^{5}}$ by cancelling the common factors.

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Step 2.1

Factor $x$ out of $x^{2} {(y^{3})}^{4}$ .

$x^{a} \cdot y^{b} = \frac{x (x {(y^{3})}^{4})}{x y^{5}}$

Step 2.2

Factor $x$ out of $x y^{5}$ .

$x^{a} \cdot y^{b} = \frac{x (x {(y^{3})}^{4})}{x (y^{5})}$

Step 2.3

Cancel the common factor.

$x^{a} \cdot y^{b} = \frac{x (x {(y^{3})}^{4})}{x y^{5}}$

Step 2.4

Rewrite the expression.

$x^{a} \cdot y^{b} = \frac{x {(y^{3})}^{4}}{y^{5}}$

Step 3

Divide each term in $x^{a} \cdot y^{b} = \frac{x {(y^{3})}^{4}}{y^{5}}$ by $y^{b}$ and simplify.

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Step 3.1

Divide each term in $x^{a} \cdot y^{b} = \frac{x {(y^{3})}^{4}}{y^{5}}$ by $y^{b}$ .

$\frac{x^{a} \cdot y^{b}}{y^{b}} = \frac{\frac{x {(y^{3})}^{4}}{y^{5}}}{y^{b}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$\frac{x^{a} \cdot y^{b}}{y^{b}} = \frac{\frac{x {(y^{3})}^{4}}{y^{5}}}{y^{b}}$

Step 3.2.1.2

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

$x^{a} = \frac{\frac{x {(y^{3})}^{4}}{y^{5}}}{y^{b}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x^{a} = \frac{x {(y^{3})}^{4}}{y^{5}} \cdot \frac{1}{y^{b}}$

Step 3.3.2

Combine.

$x^{a} = \frac{x {(y^{3})}^{4} \cdot 1}{y^{5} y^{b}}$

Step 3.3.3

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

$x^{a} = \frac{x {(y^{3})}^{4} \cdot 1}{y^{5 + b}}$

Step 3.3.4

Reduce the expression by cancelling the common factors.

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Step 3.3.4.1

Multiply $x {(y^{3})}^{4}$ by $1$ .

$x^{a} = \frac{x {(y^{3})}^{4}}{y^{5 + b}}$

Step 3.3.4.2

Multiply the exponents in ${(y^{3})}^{4}$ .

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Step 3.3.4.2.1

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

$x^{a} = \frac{x y^{3 \cdot 4}}{y^{5 + b}}$

Step 3.3.4.2.2

Multiply $3$ by $4$ .

$x^{a} = \frac{x y^{12}}{y^{5 + b}}$

Step 3.3.4.3

Cancel the common factor of $y^{12}$ and $y^{5 + b}$ .

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Step 3.3.4.3.1

Factor $y^{5 + b}$ out of $x y^{12}$ .

$x^{a} = \frac{y^{5 + b} (x y^{12 - (5 + b)})}{y^{5 + b}}$

Step 3.3.4.3.2

Cancel the common factors.

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Step 3.3.4.3.2.1

Multiply by $1$ .

$x^{a} = \frac{y^{5 + b} (x y^{12 - (5 + b)})}{y^{5 + b} \cdot 1}$

Step 3.3.4.3.2.2

Cancel the common factor.

$x^{a} = \frac{y^{5 + b} (x y^{12 - (5 + b)})}{y^{5 + b} \cdot 1}$

Step 3.3.4.3.2.3

Rewrite the expression.

$x^{a} = \frac{x y^{12 - (5 + b)}}{1}$

Step 3.3.4.3.2.4

Divide $x y^{12 - (5 + b)}$ by $1$ .

$x^{a} = x y^{12 - (5 + b)}$

Step 3.3.5

Simplify each term.

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Step 3.3.5.1

Apply the distributive property.

$x^{a} = x y^{12 - 1 \cdot 5 - b}$

Step 3.3.5.2

Multiply $- 1$ by $5$ .

$x^{a} = x y^{12 - 5 - b}$

Step 3.3.6

Subtract $5$ from $12$ .

$x^{a} = x y^{7 - b}$

Step 4

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

$\ln (x^{a}) = \ln (x y^{7 - b})$

Step 5

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

$a \ln (x) = \ln (x y^{7 - b})$

Step 6

Expand the right side.

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Step 6.1

Rewrite $\ln (x y^{7 - b})$ as $\ln (x) + \ln (y^{7 - b})$ .

$a \ln (x) = \ln (x) + \ln (y^{7 - b})$

Step 6.2

Expand $\ln (y^{7 - b})$ by moving $7 - b$ outside the logarithm.

$a \ln (x) = \ln (x) + (7 - b) \ln (y)$

Step 7

Simplify the right side.

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Step 7.1

Apply the distributive property.

$a \ln (x) = \ln (x) + 7 \ln (y) - b \ln (y)$

Step 8

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

$a \ln (x) - \ln (x) - 7 \ln (y) + b \ln (y) = 0$

Step 9

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

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Step 9.1

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

$a \ln (x) - 7 \ln (y) + b \ln (y) = \ln (x)$

Step 9.2

Add $7 \ln (y)$ to both sides of the equation.

$a \ln (x) + b \ln (y) = \ln (x) + 7 \ln (y)$

Step 9.3

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

$a \ln (x) = \ln (x) + 7 \ln (y) - b \ln (y)$

Step 10

Divide each term in $a \ln (x) = \ln (x) + 7 \ln (y) - b \ln (y)$ by $\ln (x)$ and simplify.

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Step 10.1

Divide each term in $a \ln (x) = \ln (x) + 7 \ln (y) - b \ln (y)$ by $\ln (x)$ .

$\frac{a \ln (x)}{\ln (x)} = \frac{\ln (x)}{\ln (x)} + \frac{7 \ln (y)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)}$

Step 10.2

Simplify the left side.

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Step 10.2.1

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

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Step 10.2.1.1

Cancel the common factor.

$\frac{a \ln (x)}{\ln (x)} = \frac{\ln (x)}{\ln (x)} + \frac{7 \ln (y)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)}$

Step 10.2.1.2

Divide $a$ by $1$ .

$a = \frac{\ln (x)}{\ln (x)} + \frac{7 \ln (y)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)}$

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 $\ln (x)$ .

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Step 10.3.1.1.1

Cancel the common factor.

$a = \frac{\ln (x)}{\ln (x)} + \frac{7 \ln (y)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)}$

Step 10.3.1.1.2

Rewrite the expression.

$a = 1 + \frac{7 \ln (y)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)}$

Step 10.3.1.2

Move the negative in front of the fraction.

$a = 1 + \frac{7 \ln (y)}{\ln (x)} - \frac{b \ln (y)}{\ln (x)}$