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

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

Step 1

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

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

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

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

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

Step 2

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

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

Step 3

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

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

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

$9 a b^{\frac{1}{3} \cdot 2}$

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

$9 a b^{\frac{2}{3}}$