Solve for a (6y^2)^2(2y^3)^-1=ay^b

${(6 y^{2})}^{2} {(2 y^{3})}^{- 1} = a y^{b}$

Rewrite the equation as $a y^{b} = {(6 y^{2})}^{2} {(2 y^{3})}^{- 1}$ .

$a y^{b} = {(6 y^{2})}^{2} {(2 y^{3})}^{- 1}$

Divide each term in $a y^{b} = {(6 y^{2})}^{2} {(2 y^{3})}^{- 1}$ by $y^{b}$ and simplify.

Divide each term in $a y^{b} = {(6 y^{2})}^{2} {(2 y^{3})}^{- 1}$ by $y^{b}$ .

$\frac{a y^{b}}{y^{b}} = \frac{{(6 y^{2})}^{2} {(2 y^{3})}^{- 1}}{y^{b}}$

Simplify the left side.

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

Cancel the common factor.

$\frac{a y^{b}}{y^{b}} = \frac{{(6 y^{2})}^{2} {(2 y^{3})}^{- 1}}{y^{b}}$

Divide $a$ by $1$ .

$a = \frac{{(6 y^{2})}^{2} {(2 y^{3})}^{- 1}}{y^{b}}$

Simplify the right side.

Move ${(2 y^{3})}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Multiply $y^{b}$ by $y^{3}$ by adding the exponents.

Move $y^{3}$ .

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

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

$a = \frac{{(6 y^{2})}^{2}}{y^{3 + b} \cdot 2}$

Simplify the numerator.

Apply the product rule to $6 y^{2}$ .

$a = \frac{6^{2} {(y^{2})}^{2}}{y^{3 + b} \cdot 2}$

Raise $6$ to the power of $2$ .

$a = \frac{36 {(y^{2})}^{2}}{y^{3 + b} \cdot 2}$

Multiply the exponents in ${(y^{2})}^{2}$ .

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

$a = \frac{36 y^{2 \cdot 2}}{y^{3 + b} \cdot 2}$

Multiply $2$ by $2$ .

$a = \frac{36 y^{4}}{y^{3 + b} \cdot 2}$

Simplify terms.

Cancel the common factor of $36$ and $2$ .

Factor $2$ out of $36 y^{4}$ .

$a = \frac{2 (18 y^{4})}{y^{3 + b} \cdot 2}$

Cancel the common factors.

Factor $2$ out of $y^{3 + b} \cdot 2$ .

$a = \frac{2 (18 y^{4})}{2 \cdot y^{3 + b}}$

Cancel the common factor.

$a = \frac{2 (18 y^{4})}{2 \cdot y^{3 + b}}$

Rewrite the expression.

$a = \frac{18 y^{4}}{y^{3 + b}}$

Cancel the common factor of $y^{4}$ and $y^{3 + b}$ .

Factor $y^{3 + b}$ out of $18 y^{4}$ .

$a = \frac{y^{3 + b} (18 y^{4 - (3 + b)})}{y^{3 + b}}$

Cancel the common factors.

Multiply by $1$ .

$a = \frac{y^{3 + b} (18 y^{4 - (3 + b)})}{y^{3 + b} \cdot 1}$

Cancel the common factor.

$a = \frac{y^{3 + b} (18 y^{4 - (3 + b)})}{y^{3 + b} \cdot 1}$

Rewrite the expression.

$a = \frac{18 y^{4 - (3 + b)}}{1}$

Divide $18 y^{4 - (3 + b)}$ by $1$ .

$a = 18 y^{4 - (3 + b)}$

Simplify each term.

Apply the distributive property.

$a = 18 y^{4 - 1 \cdot 3 - b}$

Multiply $- 1$ by $3$ .

$a = 18 y^{4 - 3 - b}$

Subtract $3$ from $4$ .

$a = 18 y^{1 - b}$

Solve for a (6y^2)^2(2y^3)^-1=ay^b

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