Simplify (m^(3/2)n^(-1/4))/(m^(3/4)n^(3/4))

$\frac{m^{\frac{3}{2}} n^{\frac{- 1}{4}}}{m^{\frac{3}{4}} n^{\frac{3}{4}}}$

Step 1

Move $n^{\frac{- 1}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{m^{\frac{3}{2}}}{m^{\frac{3}{4}} n^{\frac{3}{4}} n^{- \frac{- 1}{4}}}$

Step 2

Simplify the denominator.

Multiply $n^{\frac{3}{4}}$ by $n^{- \frac{- 1}{4}}$ by adding the exponents.

Move $n^{- \frac{- 1}{4}}$ .

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

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

$\frac{m^{\frac{3}{2}}}{m^{\frac{3}{4}} n^{- \frac{- 1}{4} + \frac{3}{4}}}$

Combine the numerators over the common denominator.

$\frac{m^{\frac{3}{2}}}{m^{\frac{3}{4}} n^{\frac{1 + 3}{4}}}$

Add $1$ and $3$ .

$\frac{m^{\frac{3}{2}}}{m^{\frac{3}{4}} n^{\frac{4}{4}}}$

Divide $4$ by $4$ .

$\frac{m^{\frac{3}{2}}}{m^{\frac{3}{4}} n^{1}}$

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

$\frac{m^{\frac{3}{2}}}{m^{\frac{3}{4}} n}$

Step 3

Move $m^{\frac{3}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{m^{\frac{3}{2}} m^{- \frac{3}{4}}}{n}$

Step 4

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

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

$\frac{m^{\frac{3}{2} - \frac{3}{4}}}{n}$

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

$\frac{m^{\frac{3}{2} \cdot \frac{2}{2} - \frac{3}{4}}}{n}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{m^{\frac{3 \cdot 2}{2 \cdot 2} - \frac{3}{4}}}{n}$

Multiply $2$ by $2$ .

$\frac{m^{\frac{3 \cdot 2}{4} - \frac{3}{4}}}{n}$

Combine the numerators over the common denominator.

$\frac{m^{\frac{3 \cdot 2 - 3}{4}}}{n}$

Simplify the numerator.

Multiply $3$ by $2$ .

$\frac{m^{\frac{6 - 3}{4}}}{n}$

Subtract $3$ from $6$ .

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