Simplify 2b^(1/2)(3b^(1/2)c^(1/3))^2

$2 b^{\frac{1}{2}} {(3 b^{\frac{1}{2}} c^{\frac{1}{3}})}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $3 b^{\frac{1}{2}} c^{\frac{1}{3}}$ .

$2 b^{\frac{1}{2}} ({(3 b^{\frac{1}{2}})}^{2} {(c^{\frac{1}{3}})}^{2})$

Apply the product rule to $3 b^{\frac{1}{2}}$ .

$2 b^{\frac{1}{2}} (3^{2} {(b^{\frac{1}{2}})}^{2} {(c^{\frac{1}{3}})}^{2})$

Raise $3$ to the power of $2$ .

$2 b^{\frac{1}{2}} (9 {(b^{\frac{1}{2}})}^{2} {(c^{\frac{1}{3}})}^{2})$

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

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

$2 b^{\frac{1}{2}} (9 b^{\frac{1}{2} \cdot 2} {(c^{\frac{1}{3}})}^{2})$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 b^{\frac{1}{2}} (9 b^{\frac{1}{2} \cdot 2} {(c^{\frac{1}{3}})}^{2})$

Rewrite the expression.

$2 b^{\frac{1}{2}} (9 b^{1} {(c^{\frac{1}{3}})}^{2})$

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

Move $b^{1}$ .

$2 (b^{1} b^{\frac{1}{2}}) (9 {(c^{\frac{1}{3}})}^{2})$

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

$2 b^{1 + \frac{1}{2}} (9 {(c^{\frac{1}{3}})}^{2})$

Write $1$ as a fraction with a common denominator.

$2 b^{\frac{2}{2} + \frac{1}{2}} (9 {(c^{\frac{1}{3}})}^{2})$

Combine the numerators over the common denominator.

$2 b^{\frac{2 + 1}{2}} (9 {(c^{\frac{1}{3}})}^{2})$

Add $2$ and $1$ .

$2 b^{\frac{3}{2}} (9 {(c^{\frac{1}{3}})}^{2})$

Rewrite using the commutative property of multiplication.

$2 \cdot 9 b^{\frac{3}{2}} {(c^{\frac{1}{3}})}^{2}$

Multiply $2$ by $9$ .

$18 b^{\frac{3}{2}} {(c^{\frac{1}{3}})}^{2}$

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

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

$18 b^{\frac{3}{2}} c^{\frac{1}{3} \cdot 2}$

Combine $\frac{1}{3}$ and $2$ .

$18 b^{\frac{3}{2}} c^{\frac{2}{3}}$