Simplify (4(3x^2y^4)^3)/((2x^3y^5)^4)

$\frac{4 {(3 x^{2} y^{4})}^{3}}{{(2 x^{3} y^{5})}^{4}}$

Step 1

Simplify the numerator.

Apply the product rule to $3 x^{2} y^{4}$ .

$\frac{4 {(3 x^{2})}^{3} {(y^{4})}^{3}}{{(2 x^{3} y^{5})}^{4}}$

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

$\frac{4 (3^{3} {(x^{2})}^{3}) {(y^{4})}^{3}}{{(2 x^{3} y^{5})}^{4}}$

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

$\frac{4 (27 {(x^{2})}^{3}) {(y^{4})}^{3}}{{(2 x^{3} y^{5})}^{4}}$

Multiply the exponents in ${(x^{2})}^{3}$ .

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

$\frac{4 (27 x^{2 \cdot 3}) {(y^{4})}^{3}}{{(2 x^{3} y^{5})}^{4}}$

Multiply $2$ by $3$ .

$\frac{4 (27 x^{6}) {(y^{4})}^{3}}{{(2 x^{3} y^{5})}^{4}}$

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

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

$\frac{4 (27 x^{6}) y^{4 \cdot 3}}{{(2 x^{3} y^{5})}^{4}}$

Multiply $4$ by $3$ .

$\frac{4 (27 x^{6}) y^{12}}{{(2 x^{3} y^{5})}^{4}}$

$\frac{4 \cdot 27 x^{6} y^{12}}{{(2 x^{3} y^{5})}^{4}}$

Multiply $4$ by $27$ .

$\frac{108 x^{6} y^{12}}{{(2 x^{3} y^{5})}^{4}}$

Step 2

Simplify the denominator.

Apply the product rule to $2 x^{3} y^{5}$ .

$\frac{108 x^{6} y^{12}}{{(2 x^{3})}^{4} {(y^{5})}^{4}}$

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

$\frac{108 x^{6} y^{12}}{2^{4} {(x^{3})}^{4} {(y^{5})}^{4}}$

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

$\frac{108 x^{6} y^{12}}{16 {(x^{3})}^{4} {(y^{5})}^{4}}$

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

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

$\frac{108 x^{6} y^{12}}{16 x^{3 \cdot 4} {(y^{5})}^{4}}$

Multiply $3$ by $4$ .

$\frac{108 x^{6} y^{12}}{16 x^{12} {(y^{5})}^{4}}$

Multiply the exponents in ${(y^{5})}^{4}$ .

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

$\frac{108 x^{6} y^{12}}{16 x^{12} y^{5 \cdot 4}}$

Multiply $5$ by $4$ .

$\frac{108 x^{6} y^{12}}{16 x^{12} y^{20}}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $108$ and $16$ .

Factor $4$ out of $108 x^{6} y^{12}$ .

$\frac{4 (27 x^{6} y^{12})}{16 x^{12} y^{20}}$

Cancel the common factors.

Factor $4$ out of $16 x^{12} y^{20}$ .

$\frac{4 (27 x^{6} y^{12})}{4 (4 x^{12} y^{20})}$

Cancel the common factor.

$\frac{4 (27 x^{6} y^{12})}{4 (4 x^{12} y^{20})}$

Rewrite the expression.

$\frac{27 x^{6} y^{12}}{4 x^{12} y^{20}}$

Cancel the common factor of $x^{6}$ and $x^{12}$ .

Factor $x^{6}$ out of $27 x^{6} y^{12}$ .

$\frac{x^{6} (27 y^{12})}{4 x^{12} y^{20}}$

Cancel the common factors.

Factor $x^{6}$ out of $4 x^{12} y^{20}$ .

$\frac{x^{6} (27 y^{12})}{x^{6} (4 x^{6} y^{20})}$

Cancel the common factor.

$\frac{x^{6} (27 y^{12})}{x^{6} (4 x^{6} y^{20})}$

Rewrite the expression.

$\frac{27 y^{12}}{4 x^{6} y^{20}}$

Cancel the common factor of $y^{12}$ and $y^{20}$ .

Factor $y^{12}$ out of $27 y^{12}$ .

$\frac{y^{12} \cdot 27}{4 x^{6} y^{20}}$

Cancel the common factors.

Factor $y^{12}$ out of $4 x^{6} y^{20}$ .

$\frac{y^{12} \cdot 27}{y^{12} (4 x^{6} y^{8})}$

Cancel the common factor.

$\frac{y^{12} \cdot 27}{y^{12} (4 x^{6} y^{8})}$

Rewrite the expression.

$\frac{27}{4 x^{6} y^{8}}$

Simplify (4(3x^2y^4)^3)/((2x^3y^5)^4)

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