Simplify 3m^(1/4)(mn^(1/3))^(3/2)

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

Step 1

Apply the product rule to $m n^{\frac{1}{3}}$ .

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

Step 2

Multiply $m^{\frac{1}{4}}$ by $m^{\frac{3}{2}}$ by adding the exponents.

Move $m^{\frac{3}{2}}$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $2$ .

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

Add $6$ and $1$ .

$3 m^{\frac{7}{4}} {(n^{\frac{1}{3}})}^{\frac{3}{2}}$

Step 3

Multiply the exponents in ${(n^{\frac{1}{3}})}^{\frac{3}{2}}$ .

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

$3 m^{\frac{7}{4}} n^{\frac{1}{3} \cdot \frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 m^{\frac{7}{4}} n^{\frac{1}{3} \cdot \frac{3}{2}}$

Rewrite the expression.

$3 m^{\frac{7}{4}} n^{\frac{1}{2}}$