Solve for a ((2xy)^4)/(4xy)=4x^ay^b

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

Step 1

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

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

Step 2

Divide each term in $4 x^{a} y^{b} = \frac{{(2 x y)}^{4}}{4 x y}$ by $4 y^{b}$ and simplify.

Divide each term in $4 x^{a} y^{b} = \frac{{(2 x y)}^{4}}{4 x y}$ by $4 y^{b}$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor.

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

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x^{a} = \frac{{(2 x y)}^{4}}{4 x y} \cdot \frac{1}{4 y^{b}}$

Combine.

$x^{a} = \frac{{(2 x y)}^{4} \cdot 1}{4 x y (4 y^{b})}$

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

Move $y^{b}$ .

$x^{a} = \frac{{(2 x y)}^{4} \cdot 1}{4 x (y^{b} y) \cdot 4}$

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

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

$x^{a} = \frac{{(2 x y)}^{4} \cdot 1}{4 x (y^{b} y^{1}) \cdot 4}$

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

$x^{a} = \frac{{(2 x y)}^{4} \cdot 1}{4 x y^{b + 1} \cdot 4}$

Multiply ${(2 x y)}^{4}$ by $1$ .

$x^{a} = \frac{{(2 x y)}^{4}}{4 x y^{b + 1} \cdot 4}$

Simplify the numerator.

Apply the product rule to $2 x y$ .

$x^{a} = \frac{{(2 x)}^{4} y^{4}}{4 x y^{b + 1} \cdot 4}$

Apply the product rule to $2 x$ .

$x^{a} = \frac{2^{4} x^{4} y^{4}}{4 x y^{b + 1} \cdot 4}$

Raise $2$ to the power of $4$ .

$x^{a} = \frac{16 x^{4} y^{4}}{4 x y^{b + 1} \cdot 4}$

Simplify terms.

Multiply $4$ by $4$ .

$x^{a} = \frac{16 x^{4} y^{4}}{16 x y^{b + 1}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $16$ .

Cancel the common factor.

$x^{a} = \frac{16 x^{4} y^{4}}{16 x y^{b + 1}}$

Rewrite the expression.

$x^{a} = \frac{x^{4} y^{4}}{x y^{b + 1}}$

Cancel the common factor of $x^{4}$ and $x$ .

Factor $x$ out of $x^{4} y^{4}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$x^{a} = \frac{x^{3} y^{4}}{y^{b + 1}}$

Cancel the common factor of $y^{4}$ and $y^{b + 1}$ .

Factor $y^{b + 1}$ out of $x^{3} y^{4}$ .

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

Cancel the common factors.

Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x^{3} y^{4 - (b + 1)}$ by $1$ .

$x^{a} = x^{3} y^{4 - (b + 1)}$

Simplify each term.

Apply the distributive property.

$x^{a} = x^{3} y^{4 - b - 1 \cdot 1}$

Multiply $- 1$ by $1$ .

$x^{a} = x^{3} y^{4 - b - 1}$

Subtract $1$ from $4$ .

$x^{a} = x^{3} y^{- b + 3}$

Step 3

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

$\ln (x^{a}) = \ln (x^{3} y^{- b + 3})$

Step 4

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

$a \ln (x) = \ln (x^{3} y^{- b + 3})$

Step 5

Expand the right side.

Rewrite $\ln (x^{3} y^{- b + 3})$ as $\ln (x^{3}) + \ln (y^{- b + 3})$ .

$a \ln (x) = \ln (x^{3}) + \ln (y^{- b + 3})$

Expand $\ln (x^{3})$ by moving $3$ outside the logarithm.

$a \ln (x) = 3 \ln (x) + \ln (y^{- b + 3})$

Expand $\ln (y^{- b + 3})$ by moving $- b + 3$ outside the logarithm.

$a \ln (x) = 3 \ln (x) + (- b + 3) \ln (y)$

Step 6

Simplify the right side.

Apply the distributive property.

$a \ln (x) = 3 \ln (x) - b \ln (y) + 3 \ln (y)$

Step 7

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

$a \ln (x) - 3 \ln (x) + b \ln (y) - 3 \ln (y) = 0$

Step 8

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

Add $3 \ln (x)$ to both sides of the equation.

$a \ln (x) + b \ln (y) - 3 \ln (y) = 3 \ln (x)$

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

$a \ln (x) - 3 \ln (y) = 3 \ln (x) - b \ln (y)$

Add $3 \ln (y)$ to both sides of the equation.

$a \ln (x) = 3 \ln (x) - b \ln (y) + 3 \ln (y)$

Step 9

Divide each term in $a \ln (x) = 3 \ln (x) - b \ln (y) + 3 \ln (y)$ by $\ln (x)$ and simplify.

Divide each term in $a \ln (x) = 3 \ln (x) - b \ln (y) + 3 \ln (y)$ by $\ln (x)$ .

$\frac{a \ln (x)}{\ln (x)} = \frac{3 \ln (x)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)} + \frac{3 \ln (y)}{\ln (x)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{a \ln (x)}{\ln (x)} = \frac{3 \ln (x)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)} + \frac{3 \ln (y)}{\ln (x)}$

Divide $a$ by $1$ .

$a = \frac{3 \ln (x)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)} + \frac{3 \ln (y)}{\ln (x)}$

Simplify the right side.

Simplify each term.

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

Cancel the common factor.

$a = \frac{3 \ln (x)}{\ln (x)} + \frac{- b \ln (y)}{\ln (x)} + \frac{3 \ln (y)}{\ln (x)}$

Divide $3$ by $1$ .

$a = 3 + \frac{- b \ln (y)}{\ln (x)} + \frac{3 \ln (y)}{\ln (x)}$

Move the negative in front of the fraction.

$a = 3 - \frac{b \ln (y)}{\ln (x)} + \frac{3 \ln (y)}{\ln (x)}$

Solve for a ((2xy)^4)/(4xy)=4x^ay^b

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