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

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

Step 1

Simplify the numerator.

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

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

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

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

Raise $3$ to the power of $4$ .

${(\frac{81 {(x^{4})}^{4} {(y^{2})}^{4}}{{(3 x^{5} y^{2})}^{3}})}^{2}$

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

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

${(\frac{81 x^{4 \cdot 4} {(y^{2})}^{4}}{{(3 x^{5} y^{2})}^{3}})}^{2}$

Multiply $4$ by $4$ .

${(\frac{81 x^{16} {(y^{2})}^{4}}{{(3 x^{5} y^{2})}^{3}})}^{2}$

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

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

${(\frac{81 x^{16} y^{2 \cdot 4}}{{(3 x^{5} y^{2})}^{3}})}^{2}$

Multiply $2$ by $4$ .

${(\frac{81 x^{16} y^{8}}{{(3 x^{5} y^{2})}^{3}})}^{2}$

Step 2

Simplify the denominator.

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

${(\frac{81 x^{16} y^{8}}{{(3 x^{5})}^{3} {(y^{2})}^{3}})}^{2}$

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

${(\frac{81 x^{16} y^{8}}{3^{3} {(x^{5})}^{3} {(y^{2})}^{3}})}^{2}$

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

${(\frac{81 x^{16} y^{8}}{27 {(x^{5})}^{3} {(y^{2})}^{3}})}^{2}$

Multiply the exponents in ${(x^{5})}^{3}$ .

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

${(\frac{81 x^{16} y^{8}}{27 x^{5 \cdot 3} {(y^{2})}^{3}})}^{2}$

Multiply $5$ by $3$ .

${(\frac{81 x^{16} y^{8}}{27 x^{15} {(y^{2})}^{3}})}^{2}$

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

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

${(\frac{81 x^{16} y^{8}}{27 x^{15} y^{2 \cdot 3}})}^{2}$

Multiply $2$ by $3$ .

${(\frac{81 x^{16} y^{8}}{27 x^{15} y^{6}})}^{2}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $81$ and $27$ .

Factor $27$ out of $81 x^{16} y^{8}$ .

${(\frac{27 (3 x^{16} y^{8})}{27 x^{15} y^{6}})}^{2}$

Cancel the common factors.

Factor $27$ out of $27 x^{15} y^{6}$ .

${(\frac{27 (3 x^{16} y^{8})}{27 (x^{15} y^{6})})}^{2}$

Cancel the common factor.

${(\frac{27 (3 x^{16} y^{8})}{27 (x^{15} y^{6})})}^{2}$

Rewrite the expression.

${(\frac{3 x^{16} y^{8}}{x^{15} y^{6}})}^{2}$

Cancel the common factor of $x^{16}$ and $x^{15}$ .

Factor $x^{15}$ out of $3 x^{16} y^{8}$ .

${(\frac{x^{15} (3 x y^{8})}{x^{15} y^{6}})}^{2}$

Cancel the common factors.

Factor $x^{15}$ out of $x^{15} y^{6}$ .

${(\frac{x^{15} (3 x y^{8})}{x^{15} (y^{6})})}^{2}$

Cancel the common factor.

${(\frac{x^{15} (3 x y^{8})}{x^{15} y^{6}})}^{2}$

Rewrite the expression.

${(\frac{3 x y^{8}}{y^{6}})}^{2}$

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

Factor $y^{6}$ out of $3 x y^{8}$ .

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

Cancel the common factors.

Multiply by $1$ .

${(\frac{y^{6} (3 x y^{2})}{y^{6} \cdot 1})}^{2}$

Cancel the common factor.

${(\frac{y^{6} (3 x y^{2})}{y^{6} \cdot 1})}^{2}$

Rewrite the expression.

${(\frac{3 x y^{2}}{1})}^{2}$

Divide $3 x y^{2}$ by $1$ .

${(3 x y^{2})}^{2}$

Step 4

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

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

${(3 x)}^{2} {(y^{2})}^{2}$

Apply the product rule to $3 x$ .

$3^{2} x^{2} {(y^{2})}^{2}$

Step 5

Raise $3$ to the power of $2$ .

$9 x^{2} {(y^{2})}^{2}$

Step 6

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

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

$9 x^{2} y^{2 \cdot 2}$

Multiply $2$ by $2$ .

$9 x^{2} y^{4}$

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

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