Simplify (5^(2/3)*2^(3/2))^(1/2)

${(5^{\frac{2}{3}} \cdot 2^{\frac{3}{2}})}^{\frac{1}{2}}$

Step 1

Apply the product rule to $5^{\frac{2}{3}} \cdot 2^{\frac{3}{2}}$ .

${(5^{\frac{2}{3}})}^{\frac{1}{2}} \cdot {(2^{\frac{3}{2}})}^{\frac{1}{2}}$

Step 2

Multiply the exponents in ${(5^{\frac{2}{3}})}^{\frac{1}{2}}$ .

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

$5^{\frac{2}{3} \cdot \frac{1}{2}} \cdot {(2^{\frac{3}{2}})}^{\frac{1}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$5^{\frac{2}{3} \cdot \frac{1}{2}} \cdot {(2^{\frac{3}{2}})}^{\frac{1}{2}}$

Rewrite the expression.

$5^{\frac{1}{3}} \cdot {(2^{\frac{3}{2}})}^{\frac{1}{2}}$

Step 3

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

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

$5^{\frac{1}{3}} \cdot 2^{\frac{3}{2} \cdot \frac{1}{2}}$

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

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

$5^{\frac{1}{3}} \cdot 2^{\frac{3}{2 \cdot 2}}$

Multiply $2$ by $2$ .

$5^{\frac{1}{3}} \cdot 2^{\frac{3}{4}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$5^{\frac{1}{3}} \cdot 2^{\frac{3}{4}}$

Decimal Form:

$2.87582528 \dots$