Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

Cancel the common factor of $z^{c}$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

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

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3.2

Combine.

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

Step 2.3.3

Multiply $y$ by $y^{b}$ by adding the exponents.

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Step 2.3.3.1

Move $y^{b}$ .

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

Step 2.3.3.2

Multiply $y^{b}$ by $y$ .

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

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

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

Step 2.3.4

Multiply $z$ by $z^{c}$ by adding the exponents.

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Step 2.3.4.1

Move $z^{c}$ .

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

Step 2.3.4.2

Multiply $z^{c}$ by $z$ .

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Step 2.3.4.2.1

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

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

Step 2.3.4.2.2

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

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

Step 2.3.5

Multiply ${(x^{5} y z^{4})}^{3}$ by $1$ .

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

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Apply the product rule to $x^{5} y z^{4}$ .

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

Step 2.3.6.2

Apply the product rule to $x^{5} y$ .

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

Step 2.3.6.3

Multiply the exponents in ${(x^{5})}^{3}$ .

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Step 2.3.6.3.1

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

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

Step 2.3.6.3.2

Multiply $5$ by $3$ .

$x^{a} = \frac{x^{15} y^{3} {(z^{4})}^{3}}{x^{3} y^{b + 1} z^{c + 1}}$

Step 2.3.6.4

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

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Step 2.3.6.4.1

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

$x^{a} = \frac{x^{15} y^{3} z^{4 \cdot 3}}{x^{3} y^{b + 1} z^{c + 1}}$

Step 2.3.6.4.2

Multiply $4$ by $3$ .

$x^{a} = \frac{x^{15} y^{3} z^{12}}{x^{3} y^{b + 1} z^{c + 1}}$

Step 2.3.7

Simplify terms.

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Step 2.3.7.1

Cancel the common factor of $x^{15}$ and $x^{3}$ .

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Step 2.3.7.1.1

Factor $x^{3}$ out of $x^{15} y^{3} z^{12}$ .

$x^{a} = \frac{x^{3} (x^{12} y^{3} z^{12})}{x^{3} y^{b + 1} z^{c + 1}}$

Step 2.3.7.1.2

Cancel the common factors.

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Step 2.3.7.1.2.1

Factor $x^{3}$ out of $x^{3} y^{b + 1} z^{c + 1}$ .

$x^{a} = \frac{x^{3} (x^{12} y^{3} z^{12})}{x^{3} (y^{b + 1} z^{c + 1})}$

Step 2.3.7.1.2.2

Cancel the common factor.

$x^{a} = \frac{x^{3} (x^{12} y^{3} z^{12})}{x^{3} (y^{b + 1} z^{c + 1})}$

Step 2.3.7.1.2.3

Rewrite the expression.

$x^{a} = \frac{x^{12} y^{3} z^{12}}{y^{b + 1} z^{c + 1}}$

Step 2.3.7.2

Reduce the expression by cancelling the common factors.

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Step 2.3.7.2.1

Cancel the common factor of $y^{3}$ and $y^{b + 1}$ .

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Step 2.3.7.2.1.1

Factor $y^{b + 1}$ out of $x^{12} y^{3} z^{12}$ .

$x^{a} = \frac{y^{b + 1} (x^{12} y^{3 - (b + 1)} z^{12})}{y^{b + 1} z^{c + 1}}$

Step 2.3.7.2.1.2

Cancel the common factors.

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Step 2.3.7.2.1.2.1

Factor $y^{b + 1}$ out of $y^{b + 1} z^{c + 1}$ .

$x^{a} = \frac{y^{b + 1} (x^{12} y^{3 - (b + 1)} z^{12})}{y^{b + 1} (z^{c + 1})}$

Step 2.3.7.2.1.2.2

Cancel the common factor.

$x^{a} = \frac{y^{b + 1} (x^{12} y^{3 - (b + 1)} z^{12})}{y^{b + 1} z^{c + 1}}$

Step 2.3.7.2.1.2.3

Rewrite the expression.

$x^{a} = \frac{x^{12} y^{3 - (b + 1)} z^{12}}{z^{c + 1}}$

Step 2.3.7.2.2

Cancel the common factor of $z^{12}$ and $z^{c + 1}$ .

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Step 2.3.7.2.2.1

Factor $z^{c + 1}$ out of $x^{12} y^{3 - (b + 1)} z^{12}$ .

$x^{a} = \frac{z^{c + 1} (x^{12} y^{3 - (b + 1)} z^{12 - (c + 1)})}{z^{c + 1}}$

Step 2.3.7.2.2.2

Cancel the common factors.

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Step 2.3.7.2.2.2.1

Multiply by $1$ .

$x^{a} = \frac{z^{c + 1} (x^{12} y^{3 - (b + 1)} z^{12 - (c + 1)})}{z^{c + 1} \cdot 1}$

Step 2.3.7.2.2.2.2

Cancel the common factor.

$x^{a} = \frac{z^{c + 1} (x^{12} y^{3 - (b + 1)} z^{12 - (c + 1)})}{z^{c + 1} \cdot 1}$

Step 2.3.7.2.2.2.3

Rewrite the expression.

$x^{a} = \frac{x^{12} y^{3 - (b + 1)} z^{12 - (c + 1)}}{1}$

Step 2.3.7.2.2.2.4

Divide $x^{12} y^{3 - (b + 1)} z^{12 - (c + 1)}$ by $1$ .

$x^{a} = x^{12} y^{3 - (b + 1)} z^{12 - (c + 1)}$

Step 2.3.7.3

Simplify each term.

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Step 2.3.7.3.1

Apply the distributive property.

$x^{a} = x^{12} y^{3 - b - 1 \cdot 1} z^{12 - (c + 1)}$

Step 2.3.7.3.2

Multiply $- 1$ by $1$ .

$x^{a} = x^{12} y^{3 - b - 1} z^{12 - (c + 1)}$

Step 2.3.7.4

Subtract $1$ from $3$ .

$x^{a} = x^{12} y^{- b + 2} z^{12 - (c + 1)}$

Step 2.3.7.5

Simplify each term.

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Step 2.3.7.5.1

Apply the distributive property.

$x^{a} = x^{12} y^{- b + 2} z^{12 - c - 1 \cdot 1}$

Step 2.3.7.5.2

Multiply $- 1$ by $1$ .

$x^{a} = x^{12} y^{- b + 2} z^{12 - c - 1}$

Step 2.3.7.6

Subtract $1$ from $12$ .

$x^{a} = x^{12} y^{- b + 2} z^{- c + 11}$

Step 3

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

$\ln (x^{a}) = \ln (x^{12} y^{- b + 2} z^{- c + 11})$

Step 4

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

$a \ln (x) = \ln (x^{12} y^{- b + 2} z^{- c + 11})$

Step 5

Expand the right side.

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Step 5.1

Rewrite $\ln (x^{12} y^{- b + 2} z^{- c + 11})$ as $\ln (x^{12} y^{- b + 2}) + \ln (z^{- c + 11})$ .

$a \ln (x) = \ln (x^{12} y^{- b + 2}) + \ln (z^{- c + 11})$

Step 5.2

Rewrite $\ln (x^{12} y^{- b + 2})$ as $\ln (x^{12}) + \ln (y^{- b + 2})$ .

$a \ln (x) = \ln (x^{12}) + \ln (y^{- b + 2}) + \ln (z^{- c + 11})$

Step 5.3

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

$a \ln (x) = 12 \ln (x) + \ln (y^{- b + 2}) + \ln (z^{- c + 11})$

Step 5.4

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

$a \ln (x) = 12 \ln (x) + (- b + 2) \ln (y) + \ln (z^{- c + 11})$

Step 5.5

Expand $\ln (z^{- c + 11})$ by moving $- c + 11$ outside the logarithm.

$a \ln (x) = 12 \ln (x) + (- b + 2) \ln (y) + (- c + 11) \ln (z)$

Step 6

Simplify the right side.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

$a \ln (x) = 12 \ln (x) - b \ln (y) + 2 \ln (y) + (- c + 11) \ln (z)$

Step 6.1.2

Apply the distributive property.

$a \ln (x) = 12 \ln (x) - b \ln (y) + 2 \ln (y) - c \ln (z) + 11 \ln (z)$

Step 7

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

$a \ln (x) - 12 \ln (x) + b \ln (y) - 2 \ln (y) + c \ln (z) - 11 \ln (z) = 0$

Step 8

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

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Step 8.1

Add $12 \ln (x)$ to both sides of the equation.

$a \ln (x) + b \ln (y) - 2 \ln (y) + c \ln (z) - 11 \ln (z) = 12 \ln (x)$

Step 8.2

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

$a \ln (x) - 2 \ln (y) + c \ln (z) - 11 \ln (z) = 12 \ln (x) - b \ln (y)$

Step 8.3

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

$a \ln (x) + c \ln (z) - 11 \ln (z) = 12 \ln (x) - b \ln (y) + 2 \ln (y)$

Step 8.4

Subtract $c \ln (z)$ from both sides of the equation.

$a \ln (x) - 11 \ln (z) = 12 \ln (x) - b \ln (y) + 2 \ln (y) - c \ln (z)$

Step 8.5

Add $11 \ln (z)$ to both sides of the equation.

$a \ln (x) = 12 \ln (x) - b \ln (y) + 2 \ln (y) - c \ln (z) + 11 \ln (z)$

Step 9

Divide each term in $a \ln (x) = 12 \ln (x) - b \ln (y) + 2 \ln (y) - c \ln (z) + 11 \ln (z)$ by $\ln (x)$ and simplify.

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Step 9.1

Divide each term in $a \ln (x) = 12 \ln (x) - b \ln (y) + 2 \ln (y) - c \ln (z) + 11 \ln (z)$ by $\ln (x)$ .

$\frac{a \ln (x)}{\ln (x)} = \frac{12 \ln (x)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)} + \frac{2 \ln (y)}{\ln (x)} + \frac{- c \ln (z)}{\ln (x)} + \frac{11 \ln (z)}{\ln (x)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

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

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Step 9.2.1.1

Cancel the common factor.

$\frac{a \ln (x)}{\ln (x)} = \frac{12 \ln (x)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)} + \frac{2 \ln (y)}{\ln (x)} + \frac{- c \ln (z)}{\ln (x)} + \frac{11 \ln (z)}{\ln (x)}$

Step 9.2.1.2

Divide $a$ by $1$ .

$a = \frac{12 \ln (x)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)} + \frac{2 \ln (y)}{\ln (x)} + \frac{- c \ln (z)}{\ln (x)} + \frac{11 \ln (z)}{\ln (x)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

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

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Step 9.3.1.1.1

Cancel the common factor.

$a = \frac{12 \ln (x)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)} + \frac{2 \ln (y)}{\ln (x)} + \frac{- c \ln (z)}{\ln (x)} + \frac{11 \ln (z)}{\ln (x)}$

Step 9.3.1.1.2

Divide $12$ by $1$ .

$a = 12 + \frac{- b \ln (y)}{\ln (x)} + \frac{2 \ln (y)}{\ln (x)} + \frac{- c \ln (z)}{\ln (x)} + \frac{11 \ln (z)}{\ln (x)}$

Step 9.3.1.2

Move the negative in front of the fraction.

$a = 12 - \frac{b \ln (y)}{\ln (x)} + \frac{2 \ln (y)}{\ln (x)} + \frac{- c \ln (z)}{\ln (x)} + \frac{11 \ln (z)}{\ln (x)}$

Step 9.3.1.3

Move the negative in front of the fraction.

$a = 12 - \frac{b \ln (y)}{\ln (x)} + \frac{2 \ln (y)}{\ln (x)} - \frac{c \ln (z)}{\ln (x)} + \frac{11 \ln (z)}{\ln (x)}$