Simplify the Radical Expression ((m^(1/2)n^(2/3))/(m^(2/5)n^(3/4)))^6

${(\frac{m^{\frac{1}{2}} n^{\frac{2}{3}}}{m^{\frac{2}{5}} n^{\frac{3}{4}}})}^{6}$

Step 1

Move $n^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{\frac{3}{4}} n^{- \frac{2}{3}}})}^{6}$

Step 2

Multiply $n^{\frac{3}{4}}$ by $n^{- \frac{2}{3}}$ by adding the exponents.

Move $n^{- \frac{2}{3}}$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} (n^{- \frac{2}{3}} n^{\frac{3}{4}})})}^{6}$

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

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{- \frac{2}{3} + \frac{3}{4}}})}^{6}$

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{- \frac{2}{3} \cdot \frac{4}{4} + \frac{3}{4}}})}^{6}$

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{- \frac{2}{3} \cdot \frac{4}{4} + \frac{3}{4} \cdot \frac{3}{3}}})}^{6}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{3}$ by $\frac{4}{4}$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{- \frac{2 \cdot 4}{3 \cdot 4} + \frac{3}{4} \cdot \frac{3}{3}}})}^{6}$

Multiply $3$ by $4$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{- \frac{2 \cdot 4}{12} + \frac{3}{4} \cdot \frac{3}{3}}})}^{6}$

Multiply $\frac{3}{4}$ by $\frac{3}{3}$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{- \frac{2 \cdot 4}{12} + \frac{3 \cdot 3}{4 \cdot 3}}})}^{6}$

Multiply $4$ by $3$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{- \frac{2 \cdot 4}{12} + \frac{3 \cdot 3}{12}}})}^{6}$

Combine the numerators over the common denominator.

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{\frac{- 2 \cdot 4 + 3 \cdot 3}{12}}})}^{6}$

Simplify the numerator.

Multiply $- 2$ by $4$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{\frac{- 8 + 3 \cdot 3}{12}}})}^{6}$

Multiply $3$ by $3$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{\frac{- 8 + 9}{12}}})}^{6}$

Add $- 8$ and $9$ .

${(\frac{m^{\frac{1}{2}}}{m^{\frac{2}{5}} n^{\frac{1}{12}}})}^{6}$

Step 3

Move $m^{\frac{2}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

${(\frac{m^{\frac{1}{2}} m^{- \frac{2}{5}}}{n^{\frac{1}{12}}})}^{6}$

Step 4

Multiply $m^{\frac{1}{2}}$ by $m^{- \frac{2}{5}}$ by adding the exponents.

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

${(\frac{m^{\frac{1}{2} - \frac{2}{5}}}{n^{\frac{1}{12}}})}^{6}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(\frac{m^{\frac{1}{2} \cdot \frac{5}{5} - \frac{2}{5}}}{n^{\frac{1}{12}}})}^{6}$

To write $- \frac{2}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{m^{\frac{1}{2} \cdot \frac{5}{5} - \frac{2}{5} \cdot \frac{2}{2}}}{n^{\frac{1}{12}}})}^{6}$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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

${(\frac{m^{\frac{5}{2 \cdot 5} - \frac{2}{5} \cdot \frac{2}{2}}}{n^{\frac{1}{12}}})}^{6}$

Multiply $2$ by $5$ .

${(\frac{m^{\frac{5}{10} - \frac{2}{5} \cdot \frac{2}{2}}}{n^{\frac{1}{12}}})}^{6}$

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

${(\frac{m^{\frac{5}{10} - \frac{2 \cdot 2}{5 \cdot 2}}}{n^{\frac{1}{12}}})}^{6}$

Multiply $5$ by $2$ .

${(\frac{m^{\frac{5}{10} - \frac{2 \cdot 2}{10}}}{n^{\frac{1}{12}}})}^{6}$

Combine the numerators over the common denominator.

${(\frac{m^{\frac{5 - 2 \cdot 2}{10}}}{n^{\frac{1}{12}}})}^{6}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

${(\frac{m^{\frac{5 - 4}{10}}}{n^{\frac{1}{12}}})}^{6}$

Subtract $4$ from $5$ .

${(\frac{m^{\frac{1}{10}}}{n^{\frac{1}{12}}})}^{6}$

Step 5

Apply the product rule to $\frac{m^{\frac{1}{10}}}{n^{\frac{1}{12}}}$ .

$\frac{{(m^{\frac{1}{10}})}^{6}}{{(n^{\frac{1}{12}})}^{6}}$

Step 6

Multiply the exponents in ${(m^{\frac{1}{10}})}^{6}$ .

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

$\frac{m^{\frac{1}{10} \cdot 6}}{{(n^{\frac{1}{12}})}^{6}}$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$\frac{m^{\frac{1}{2 (5)} \cdot 6}}{{(n^{\frac{1}{12}})}^{6}}$

Factor $2$ out of $6$ .

$\frac{m^{\frac{1}{2 \cdot 5} \cdot (2 \cdot 3)}}{{(n^{\frac{1}{12}})}^{6}}$

Cancel the common factor.

$\frac{m^{\frac{1}{2 \cdot 5} \cdot (2 \cdot 3)}}{{(n^{\frac{1}{12}})}^{6}}$

Rewrite the expression.

$\frac{m^{\frac{1}{5} \cdot 3}}{{(n^{\frac{1}{12}})}^{6}}$

Combine $\frac{1}{5}$ and $3$ .

$\frac{m^{\frac{3}{5}}}{{(n^{\frac{1}{12}})}^{6}}$

Step 7

Multiply the exponents in ${(n^{\frac{1}{12}})}^{6}$ .

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

$\frac{m^{\frac{3}{5}}}{n^{\frac{1}{12} \cdot 6}}$

Cancel the common factor of $6$ .

Factor $6$ out of $12$ .

$\frac{m^{\frac{3}{5}}}{n^{\frac{1}{6 (2)} \cdot 6}}$

Cancel the common factor.

$\frac{m^{\frac{3}{5}}}{n^{\frac{1}{6 \cdot 2} \cdot 6}}$

Rewrite the expression.

$\frac{m^{\frac{3}{5}}}{n^{\frac{1}{2}}}$

Simplify the Radical Expression ((m^(1/2)n^(2/3))/(m^(2/5)n^(3/4)))^6

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