Simplify the Radical Expression (27m^12n^3)^(1/3)

${(27 m^{12} n^{3})}^{\frac{1}{3}}$

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

Apply the product rule to $27 m^{12} n^{3}$ .

${(27 m^{12})}^{\frac{1}{3}} {(n^{3})}^{\frac{1}{3}}$

Apply the product rule to $27 m^{12}$ .

$27^{\frac{1}{3}} {(m^{12})}^{\frac{1}{3}} {(n^{3})}^{\frac{1}{3}}$

Rewrite $27$ as $3^{3}$ .

${(3^{3})}^{\frac{1}{3}} {(m^{12})}^{\frac{1}{3}} {(n^{3})}^{\frac{1}{3}}$

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

$3^{3 (\frac{1}{3})} {(m^{12})}^{\frac{1}{3}} {(n^{3})}^{\frac{1}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$3^{3 (\frac{1}{3})} {(m^{12})}^{\frac{1}{3}} {(n^{3})}^{\frac{1}{3}}$

Rewrite the expression.

$3^{1} {(m^{12})}^{\frac{1}{3}} {(n^{3})}^{\frac{1}{3}}$

Evaluate the exponent.

$3 {(m^{12})}^{\frac{1}{3}} {(n^{3})}^{\frac{1}{3}}$

Multiply the exponents in ${(m^{12})}^{\frac{1}{3}}$ .

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

$3 m^{12 (\frac{1}{3})} {(n^{3})}^{\frac{1}{3}}$

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

$3 m^{3 (4) \frac{1}{3}} {(n^{3})}^{\frac{1}{3}}$

Cancel the common factor.

$3 m^{3 \cdot 4 \frac{1}{3}} {(n^{3})}^{\frac{1}{3}}$

Rewrite the expression.

$3 m^{4} {(n^{3})}^{\frac{1}{3}}$

Multiply the exponents in ${(n^{3})}^{\frac{1}{3}}$ .

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

$3 m^{4} n^{3 (\frac{1}{3})}$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 m^{4} n^{3 (\frac{1}{3})}$

Rewrite the expression.

$3 m^{4} n^{1}$

Simplify.

$3 m^{4} n$