Simplify ((m^5n^5)^(1/6))/(3(mn)^(-1/6))

$\frac{{(m^{5} n^{5})}^{\frac{1}{6}}}{3 {(m n)}^{- \frac{1}{6}}}$

Step 1

Move ${(m n)}^{- \frac{1}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(m^{5} n^{5})}^{\frac{1}{6}} {(m n)}^{\frac{1}{6}}}{3}$

Step 2

Simplify the numerator.

Apply the product rule to $m^{5} n^{5}$ .

$\frac{{(m^{5})}^{\frac{1}{6}} {(n^{5})}^{\frac{1}{6}} {(m n)}^{\frac{1}{6}}}{3}$

Multiply the exponents in ${(m^{5})}^{\frac{1}{6}}$ .

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

$\frac{m^{5 (\frac{1}{6})} {(n^{5})}^{\frac{1}{6}} {(m n)}^{\frac{1}{6}}}{3}$

Combine $5$ and $\frac{1}{6}$ .

$\frac{m^{\frac{5}{6}} {(n^{5})}^{\frac{1}{6}} {(m n)}^{\frac{1}{6}}}{3}$

Multiply the exponents in ${(n^{5})}^{\frac{1}{6}}$ .

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

$\frac{m^{\frac{5}{6}} n^{5 (\frac{1}{6})} {(m n)}^{\frac{1}{6}}}{3}$

Combine $5$ and $\frac{1}{6}$ .

$\frac{m^{\frac{5}{6}} n^{\frac{5}{6}} {(m n)}^{\frac{1}{6}}}{3}$

Apply the product rule to $m n$ .

$\frac{m^{\frac{5}{6}} n^{\frac{5}{6}} m^{\frac{1}{6}} n^{\frac{1}{6}}}{3}$

Combine exponents.

Multiply $m^{\frac{5}{6}}$ by $m^{\frac{1}{6}}$ by adding the exponents.

Move $m^{\frac{1}{6}}$ .

$\frac{m^{\frac{1}{6}} m^{\frac{5}{6}} n^{\frac{5}{6}} n^{\frac{1}{6}}}{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{m^{\frac{1}{6} + \frac{5}{6}} n^{\frac{5}{6}} n^{\frac{1}{6}}}{3}$

Combine the numerators over the common denominator.

$\frac{m^{\frac{1 + 5}{6}} n^{\frac{5}{6}} n^{\frac{1}{6}}}{3}$

Add $1$ and $5$ .

$\frac{m^{\frac{6}{6}} n^{\frac{5}{6}} n^{\frac{1}{6}}}{3}$

Divide $6$ by $6$ .

$\frac{m^{1} n^{\frac{5}{6}} n^{\frac{1}{6}}}{3}$

Simplify $m^{1} n^{\frac{5}{6}}$ .

$\frac{m n^{\frac{5}{6}} n^{\frac{1}{6}}}{3}$

Multiply $n^{\frac{5}{6}}$ by $n^{\frac{1}{6}}$ by adding the exponents.

Move $n^{\frac{1}{6}}$ .

$\frac{m (n^{\frac{1}{6}} n^{\frac{5}{6}})}{3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{m n^{\frac{1}{6} + \frac{5}{6}}}{3}$

Combine the numerators over the common denominator.

$\frac{m n^{\frac{1 + 5}{6}}}{3}$

Add $1$ and $5$ .

$\frac{m n^{\frac{6}{6}}}{3}$

Divide $6$ by $6$ .

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

Simplify $m n^{1}$ .

$\frac{m n}{3}$

Simplify ((m^5n^5)^(1/6))/(3(mn)^(-1/6))

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