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

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

Step 1

Multiply ${(2 a^{2})}^{\frac{1}{2}}$ by ${(2 a^{2})}^{\frac{1}{3}}$ by adding the exponents.

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

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

${(2 a^{2})}^{\frac{3}{6} + \frac{1}{3} \cdot \frac{2}{2}}$

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

${(2 a^{2})}^{\frac{3}{6} + \frac{2}{3 \cdot 2}}$

Multiply $3$ by $2$ .

${(2 a^{2})}^{\frac{3}{6} + \frac{2}{6}}$

Combine the numerators over the common denominator.

${(2 a^{2})}^{\frac{3 + 2}{6}}$

Add $3$ and $2$ .

${(2 a^{2})}^{\frac{5}{6}}$

Step 2

Apply the product rule to $2 a^{2}$ .

$2^{\frac{5}{6}} {(a^{2})}^{\frac{5}{6}}$

Step 3

Multiply the exponents in ${(a^{2})}^{\frac{5}{6}}$ .

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

$2^{\frac{5}{6}} a^{2 (\frac{5}{6})}$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$2^{\frac{5}{6}} a^{2 \frac{5}{2 (3)}}$

Cancel the common factor.

$2^{\frac{5}{6}} a^{2 \frac{5}{2 \cdot 3}}$

Rewrite the expression.

$2^{\frac{5}{6}} a^{\frac{5}{3}}$