Solve for b (5x^7y^2)(-4x^4y^5)=-20x^ay^b

$(5 x^{7} y^{2}) (- 4 x^{4} y^{5}) = - 20 x^{a} y^{b}$

Rewrite the equation as $- 20 x^{a} y^{b} = 5 x^{7} y^{2} (- 4 x^{4} y^{5})$ .

$- 20 x^{a} y^{b} = 5 x^{7} y^{2} (- 4 x^{4} y^{5})$

Simplify.

Multiply $- 4$ by $5$ .

$- 20 x^{a} y^{b} = - 20 x^{7} y^{2} (x^{4} y^{5})$

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

$- 20 x^{a} y^{b} = - 20 x^{4 + 7} y^{2} y^{5}$

Add $4$ and $7$ .

$- 20 x^{a} y^{b} = - 20 x^{11} y^{2} y^{5}$

Multiply $y^{2}$ by $y^{5}$ by adding the exponents.

Move $y^{5}$ .

$- 20 x^{a} y^{b} = - 20 x^{11} (y^{5} y^{2})$

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

$- 20 x^{a} y^{b} = - 20 x^{11} y^{5 + 2}$

Add $5$ and $2$ .

$- 20 x^{a} y^{b} = - 20 x^{11} y^{7}$

Divide each term in $- 20 x^{a} y^{b} = - 20 x^{11} y^{7}$ by $- 20 x^{a}$ and simplify.

Divide each term in $- 20 x^{a} y^{b} = - 20 x^{11} y^{7}$ by $- 20 x^{a}$ .

$\frac{- 20 x^{a} y^{b}}{- 20 x^{a}} = \frac{- 20 x^{11} y^{7}}{- 20 x^{a}}$

Simplify the left side.

Cancel the common factor of $- 20$ .

Cancel the common factor.

$\frac{- 20 x^{a} y^{b}}{- 20 x^{a}} = \frac{- 20 x^{11} y^{7}}{- 20 x^{a}}$

Rewrite the expression.

$\frac{x^{a} y^{b}}{x^{a}} = \frac{- 20 x^{11} y^{7}}{- 20 x^{a}}$

Cancel the common factor of $x^{a}$ .

Cancel the common factor.

$\frac{x^{a} y^{b}}{x^{a}} = \frac{- 20 x^{11} y^{7}}{- 20 x^{a}}$

Divide $y^{b}$ by $1$ .

$y^{b} = \frac{- 20 x^{11} y^{7}}{- 20 x^{a}}$

Simplify the right side.

Cancel the common factor of $- 20$ .

Cancel the common factor.

$y^{b} = \frac{- 20 x^{11} y^{7}}{- 20 x^{a}}$

Rewrite the expression.

$y^{b} = \frac{x^{11} y^{7}}{x^{a}}$

Cancel the common factor of $x^{11}$ and $x^{a}$ .

Factor $x^{a}$ out of $x^{11} y^{7}$ .

$y^{b} = \frac{x^{a} (x^{11 - a} y^{7})}{x^{a}}$

Cancel the common factors.

Multiply by $1$ .

$y^{b} = \frac{x^{a} (x^{11 - a} y^{7})}{x^{a} \cdot 1}$

Cancel the common factor.

$y^{b} = \frac{x^{a} (x^{11 - a} y^{7})}{x^{a} \cdot 1}$

Rewrite the expression.

$y^{b} = \frac{x^{11 - a} y^{7}}{1}$

Divide $x^{11 - a} y^{7}$ by $1$ .

$y^{b} = x^{11 - a} y^{7}$

Reorder factors in $x^{11 - a} y^{7}$ .

$y^{b} = y^{7} x^{11 - a}$

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

$\ln (y^{b}) = \ln (y^{7} x^{11 - a})$

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

$b \ln (y) = \ln (y^{7} x^{11 - a})$

Expand the right side.

Rewrite $\ln (y^{7} x^{11 - a})$ as $\ln (y^{7}) + \ln (x^{11 - a})$ .

$b \ln (y) = \ln (y^{7}) + \ln (x^{11 - a})$

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

$b \ln (y) = 7 \ln (y) + \ln (x^{11 - a})$

Expand $\ln (x^{11 - a})$ by moving $11 - a$ outside the logarithm.

$b \ln (y) = 7 \ln (y) + (11 - a) \ln (x)$

Simplify the right side.

Apply the distributive property.

$b \ln (y) = 7 \ln (y) + 11 \ln (x) - a \ln (x)$

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

$b \ln (y) - 7 \ln (y) - 11 \ln (x) + a \ln (x) = 0$

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

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

$b \ln (y) - 11 \ln (x) + a \ln (x) = 7 \ln (y)$

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

$b \ln (y) + a \ln (x) = 7 \ln (y) + 11 \ln (x)$

Subtract $a \ln (x)$ from both sides of the equation.

$b \ln (y) = 7 \ln (y) + 11 \ln (x) - a \ln (x)$

Divide each term in $b \ln (y) = 7 \ln (y) + 11 \ln (x) - a \ln (x)$ by $\ln (y)$ and simplify.

Divide each term in $b \ln (y) = 7 \ln (y) + 11 \ln (x) - a \ln (x)$ by $\ln (y)$ .

$\frac{b \ln (y)}{\ln (y)} = \frac{7 \ln (y)}{\ln (y)} + \frac{11 \ln (x)}{\ln (y)} + \frac{- a \ln (x)}{\ln (y)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{b \ln (y)}{\ln (y)} = \frac{7 \ln (y)}{\ln (y)} + \frac{11 \ln (x)}{\ln (y)} + \frac{- a \ln (x)}{\ln (y)}$

Divide $b$ by $1$ .

$b = \frac{7 \ln (y)}{\ln (y)} + \frac{11 \ln (x)}{\ln (y)} + \frac{- a \ln (x)}{\ln (y)}$

Simplify the right side.

Simplify each term.

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

Cancel the common factor.

$b = \frac{7 \ln (y)}{\ln (y)} + \frac{11 \ln (x)}{\ln (y)} + \frac{- a \ln (x)}{\ln (y)}$

Divide $7$ by $1$ .

$b = 7 + \frac{11 \ln (x)}{\ln (y)} + \frac{- a \ln (x)}{\ln (y)}$

Move the negative in front of the fraction.

$b = 7 + \frac{11 \ln (x)}{\ln (y)} - \frac{a \ln (x)}{\ln (y)}$

Solve for b (5x^7y^2)(-4x^4y^5)=-20x^ay^b

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