Simplify ((p^12g^(3/2))/64)^(5/6)

${(\frac{p^{12} g^{\frac{3}{2}}}{64})}^{\frac{5}{6}}$

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

Apply the product rule to $\frac{p^{12} g^{\frac{3}{2}}}{64}$ .

$\frac{{(p^{12} g^{\frac{3}{2}})}^{\frac{5}{6}}}{64^{\frac{5}{6}}}$

Apply the product rule to $p^{12} g^{\frac{3}{2}}$ .

$\frac{{(p^{12})}^{\frac{5}{6}} {(g^{\frac{3}{2}})}^{\frac{5}{6}}}{64^{\frac{5}{6}}}$

Simplify the numerator.

Multiply the exponents in ${(p^{12})}^{\frac{5}{6}}$ .

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

$\frac{p^{12 (\frac{5}{6})} {(g^{\frac{3}{2}})}^{\frac{5}{6}}}{64^{\frac{5}{6}}}$

Cancel the common factor of $6$ .

Factor $6$ out of $12$ .

$\frac{p^{6 (2) \frac{5}{6}} {(g^{\frac{3}{2}})}^{\frac{5}{6}}}{64^{\frac{5}{6}}}$

Cancel the common factor.

$\frac{p^{6 \cdot 2 \frac{5}{6}} {(g^{\frac{3}{2}})}^{\frac{5}{6}}}{64^{\frac{5}{6}}}$

Rewrite the expression.

$\frac{p^{2 \cdot 5} {(g^{\frac{3}{2}})}^{\frac{5}{6}}}{64^{\frac{5}{6}}}$

Multiply $2$ by $5$ .

$\frac{p^{10} {(g^{\frac{3}{2}})}^{\frac{5}{6}}}{64^{\frac{5}{6}}}$

Multiply the exponents in ${(g^{\frac{3}{2}})}^{\frac{5}{6}}$ .

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

$\frac{p^{10} g^{\frac{3}{2} \cdot \frac{5}{6}}}{64^{\frac{5}{6}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$\frac{p^{10} g^{\frac{3}{2} \cdot \frac{5}{3 (2)}}}{64^{\frac{5}{6}}}$

Cancel the common factor.

$\frac{p^{10} g^{\frac{3}{2} \cdot \frac{5}{3 \cdot 2}}}{64^{\frac{5}{6}}}$

Rewrite the expression.

$\frac{p^{10} g^{\frac{1}{2} \cdot \frac{5}{2}}}{64^{\frac{5}{6}}}$

Multiply $\frac{1}{2}$ by $\frac{5}{2}$ .

$\frac{p^{10} g^{\frac{5}{2 \cdot 2}}}{64^{\frac{5}{6}}}$

Multiply $2$ by $2$ .

$\frac{p^{10} g^{\frac{5}{4}}}{64^{\frac{5}{6}}}$

Simplify the denominator.

Rewrite $64$ as $2^{6}$ .

$\frac{p^{10} g^{\frac{5}{4}}}{{(2^{6})}^{\frac{5}{6}}}$

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

$\frac{p^{10} g^{\frac{5}{4}}}{2^{6 (\frac{5}{6})}}$

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{p^{10} g^{\frac{5}{4}}}{2^{6 (\frac{5}{6})}}$

Rewrite the expression.

$\frac{p^{10} g^{\frac{5}{4}}}{2^{5}}$

Raise $2$ to the power of $5$ .

$\frac{p^{10} g^{\frac{5}{4}}}{32}$