Simplify (3x^(1/2))(4x^(2/3))

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

Rewrite using the commutative property of multiplication.

$3 \cdot 4 x^{\frac{1}{2}} x^{\frac{2}{3}}$

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

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

$3 \cdot 4 (x^{\frac{2}{3}} x^{\frac{1}{2}})$

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

$3 \cdot 4 x^{\frac{2}{3} + \frac{1}{2}}$

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

$3 \cdot 4 x^{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2}}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 \cdot 4 x^{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3}}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$3 \cdot 4 x^{\frac{2 \cdot 2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3}}$

Multiply $3$ by $2$ .

$3 \cdot 4 x^{\frac{2 \cdot 2}{6} + \frac{1}{2} \cdot \frac{3}{3}}$

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$3 \cdot 4 x^{\frac{2 \cdot 2}{6} + \frac{3}{2 \cdot 3}}$

Multiply $2$ by $3$ .

$3 \cdot 4 x^{\frac{2 \cdot 2}{6} + \frac{3}{6}}$

Combine the numerators over the common denominator.

$3 \cdot 4 x^{\frac{2 \cdot 2 + 3}{6}}$

Simplify the numerator.

Multiply $2$ by $2$ .

$3 \cdot 4 x^{\frac{4 + 3}{6}}$

Add $4$ and $3$ .

$3 \cdot 4 x^{\frac{7}{6}}$

Multiply $3$ by $4$ .

$12 x^{\frac{7}{6}}$