Simplify the Radical Expression ((16m^9n^3)/(2^6))^(1/3)

${(\frac{16 m^{9} n^{3}}{2^{6}})}^{\frac{1}{3}}$

Step 1

Raise $2$ to the power of $6$ .

${(\frac{16 m^{9} n^{3}}{64})}^{\frac{1}{3}}$

Step 2

Cancel the common factor of $16$ and $64$ .

Factor $16$ out of $16 m^{9} n^{3}$ .

${(\frac{16 (m^{9} n^{3})}{64})}^{\frac{1}{3}}$

Cancel the common factors.

Factor $16$ out of $64$ .

${(\frac{16 (m^{9} n^{3})}{16 \cdot 4})}^{\frac{1}{3}}$

Cancel the common factor.

${(\frac{16 (m^{9} n^{3})}{16 \cdot 4})}^{\frac{1}{3}}$

Rewrite the expression.

${(\frac{m^{9} n^{3}}{4})}^{\frac{1}{3}}$

Step 3

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

Apply the product rule to $\frac{m^{9} n^{3}}{4}$ .

$\frac{{(m^{9} n^{3})}^{\frac{1}{3}}}{4^{\frac{1}{3}}}$

Apply the product rule to $m^{9} n^{3}$ .

$\frac{{(m^{9})}^{\frac{1}{3}} {(n^{3})}^{\frac{1}{3}}}{4^{\frac{1}{3}}}$

Step 4

Simplify the numerator.

Multiply the exponents in ${(m^{9})}^{\frac{1}{3}}$ .

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

$\frac{m^{9 (\frac{1}{3})} {(n^{3})}^{\frac{1}{3}}}{4^{\frac{1}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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