Simplify ((3a^2)/b)^6((2b^2)/a)^4

${(\frac{3 a^{2}}{b})}^{6} {(\frac{2 b^{2}}{a})}^{4}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{3 a^{2}}{b}$ .

$\frac{{(3 a^{2})}^{6}}{b^{6}} {(\frac{2 b^{2}}{a})}^{4}$

Apply the product rule to $3 a^{2}$ .

$\frac{3^{6} {(a^{2})}^{6}}{b^{6}} {(\frac{2 b^{2}}{a})}^{4}$

Step 2

Simplify the numerator.

Raise $3$ to the power of $6$ .

$\frac{729 {(a^{2})}^{6}}{b^{6}} {(\frac{2 b^{2}}{a})}^{4}$

Multiply the exponents in ${(a^{2})}^{6}$ .

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

$\frac{729 a^{2 \cdot 6}}{b^{6}} {(\frac{2 b^{2}}{a})}^{4}$

Multiply $2$ by $6$ .

$\frac{729 a^{12}}{b^{6}} {(\frac{2 b^{2}}{a})}^{4}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{2 b^{2}}{a}$ .

$\frac{729 a^{12}}{b^{6}} \cdot \frac{{(2 b^{2})}^{4}}{a^{4}}$

Apply the product rule to $2 b^{2}$ .

$\frac{729 a^{12}}{b^{6}} \cdot \frac{2^{4} {(b^{2})}^{4}}{a^{4}}$

Step 4

Simplify terms.

Combine.

$\frac{729 a^{12} (2^{4} {(b^{2})}^{4})}{b^{6} a^{4}}$

Cancel the common factor of $a^{12}$ and $a^{4}$ .

Factor $a^{4}$ out of $729 a^{12} (2^{4} {(b^{2})}^{4})$ .

$\frac{a^{4} (729 a^{8} (2^{4} {(b^{2})}^{4}))}{b^{6} a^{4}}$

Cancel the common factors.

Factor $a^{4}$ out of $b^{6} a^{4}$ .

$\frac{a^{4} (729 a^{8} (2^{4} {(b^{2})}^{4}))}{a^{4} b^{6}}$

Cancel the common factor.

$\frac{a^{4} (729 a^{8} (2^{4} {(b^{2})}^{4}))}{a^{4} b^{6}}$

Rewrite the expression.

$\frac{729 a^{8} (2^{4} {(b^{2})}^{4})}{b^{6}}$

Step 5

Simplify the numerator.

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

$\frac{729 a^{8} \cdot 16 {(b^{2})}^{4}}{b^{6}}$

Multiply the exponents in ${(b^{2})}^{4}$ .

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

$\frac{729 a^{8} \cdot 16 b^{2 \cdot 4}}{b^{6}}$

Multiply $2$ by $4$ .

$\frac{729 a^{8} \cdot 16 b^{8}}{b^{6}}$

Multiply $16$ by $729$ .

$\frac{11664 a^{8} b^{8}}{b^{6}}$

Step 6

Cancel the common factor of $b^{8}$ and $b^{6}$ .

Factor $b^{6}$ out of $11664 a^{8} b^{8}$ .

$\frac{b^{6} (11664 a^{8} b^{2})}{b^{6}}$

Cancel the common factors.

Multiply by $1$ .

$\frac{b^{6} (11664 a^{8} b^{2})}{b^{6} \cdot 1}$

Cancel the common factor.

$\frac{b^{6} (11664 a^{8} b^{2})}{b^{6} \cdot 1}$

Rewrite the expression.

$\frac{11664 a^{8} b^{2}}{1}$

Divide $11664 a^{8} b^{2}$ by $1$ .

$11664 a^{8} b^{2}$

Simplify ((3a^2)/b)^6((2b^2)/a)^4

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