Simplify ((3gh^4)/(g^5h^2))^4

${(\frac{3 g h^{4}}{g^{5} h^{2}})}^{4}$

Step 1

Cancel the common factor of $g$ and $g^{5}$ .

Factor $g$ out of $3 g h^{4}$ .

${(\frac{g (3 h^{4})}{g^{5} h^{2}})}^{4}$

Cancel the common factors.

Factor $g$ out of $g^{5} h^{2}$ .

${(\frac{g (3 h^{4})}{g (g^{4} h^{2})})}^{4}$

Cancel the common factor.

${(\frac{g (3 h^{4})}{g (g^{4} h^{2})})}^{4}$

Rewrite the expression.

${(\frac{3 h^{4}}{g^{4} h^{2}})}^{4}$

Step 2

Cancel the common factor of $h^{4}$ and $h^{2}$ .

Factor $h^{2}$ out of $3 h^{4}$ .

${(\frac{h^{2} (3 h^{2})}{g^{4} h^{2}})}^{4}$

Cancel the common factors.

Factor $h^{2}$ out of $g^{4} h^{2}$ .

${(\frac{h^{2} (3 h^{2})}{h^{2} g^{4}})}^{4}$

Cancel the common factor.

${(\frac{h^{2} (3 h^{2})}{h^{2} g^{4}})}^{4}$

Rewrite the expression.

${(\frac{3 h^{2}}{g^{4}})}^{4}$

Step 3

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

Apply the product rule to $\frac{3 h^{2}}{g^{4}}$ .

$\frac{{(3 h^{2})}^{4}}{{(g^{4})}^{4}}$

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

$\frac{3^{4} {(h^{2})}^{4}}{{(g^{4})}^{4}}$

Step 4

Simplify the numerator.

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

$\frac{81 {(h^{2})}^{4}}{{(g^{4})}^{4}}$

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

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

$\frac{81 h^{2 \cdot 4}}{{(g^{4})}^{4}}$

Multiply $2$ by $4$ .

$\frac{81 h^{8}}{{(g^{4})}^{4}}$

Step 5

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

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

$\frac{81 h^{8}}{g^{4 \cdot 4}}$

Multiply $4$ by $4$ .

$\frac{81 h^{8}}{g^{16}}$