Simplify ((6*4^n)/(4^(2n+1)+2^(4n+1)))^(n^-1)

${(\frac{6 \times 4^{n}}{4^{2 n + 1} + 2^{4 n + 1}})}^{n^{- 1}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{6 \cdot 4^{n}}{4^{2 n + 1} + 2^{4 n + 1}})}^{\frac{1}{n}}$

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

Apply the product rule to $\frac{6 \cdot 4^{n}}{4^{2 n + 1} + 2^{4 n + 1}}$ .

$\frac{{(6 \cdot 4^{n})}^{\frac{1}{n}}}{{(4^{2 n + 1} + 2^{4 n + 1})}^{\frac{1}{n}}}$

Apply the product rule to $6 \cdot 4^{n}$ .

$\frac{6^{\frac{1}{n}} \cdot {(4^{n})}^{\frac{1}{n}}}{{(4^{2 n + 1} + 2^{4 n + 1})}^{\frac{1}{n}}}$

Simplify the numerator.

Multiply the exponents in ${(4^{n})}^{\frac{1}{n}}$ .

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

$\frac{6^{\frac{1}{n}} \cdot 4^{n \frac{1}{n}}}{{(4^{2 n + 1} + 2^{4 n + 1})}^{\frac{1}{n}}}$

Cancel the common factor of $n$ .

Cancel the common factor.

$\frac{6^{\frac{1}{n}} \cdot 4^{n \frac{1}{n}}}{{(4^{2 n + 1} + 2^{4 n + 1})}^{\frac{1}{n}}}$

Rewrite the expression.

$\frac{6^{\frac{1}{n}} \cdot 4^{1}}{{(4^{2 n + 1} + 2^{4 n + 1})}^{\frac{1}{n}}}$

Evaluate the exponent.

$\frac{6^{\frac{1}{n}} \cdot 4}{{(4^{2 n + 1} + 2^{4 n + 1})}^{\frac{1}{n}}}$

Move $4$ to the left of $6^{\frac{1}{n}}$ .

$\frac{4 \cdot 6^{\frac{1}{n}}}{{(4^{2 n + 1} + 2^{4 n + 1})}^{\frac{1}{n}}}$