Evaluate square root of 16x^2*x^(2/3)+ cube root of 8x^5

$\sqrt{16 x^{2}} \cdot x^{\frac{2}{3}} + \sqrt[3]{8 x^{5}}$

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$\sqrt{{(4 x)}^{2}} \cdot x^{\frac{2}{3}} + \sqrt[3]{8 x^{5}}$

Pull terms out from under the radical, assuming positive real numbers.

$4 x \cdot x^{\frac{2}{3}} + \sqrt[3]{8 x^{5}}$

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

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

$4 (x^{\frac{2}{3}} x) + \sqrt[3]{8 x^{5}}$

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

Raise $x$ to the power of $1$ .

$4 (x^{\frac{2}{3}} x^{1}) + \sqrt[3]{8 x^{5}}$

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

$4 x^{\frac{2}{3} + 1} + \sqrt[3]{8 x^{5}}$

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

$4 x^{\frac{2}{3} + \frac{3}{3}} + \sqrt[3]{8 x^{5}}$

Combine the numerators over the common denominator.

$4 x^{\frac{2 + 3}{3}} + \sqrt[3]{8 x^{5}}$

Add $2$ and $3$ .

$4 x^{\frac{5}{3}} + \sqrt[3]{8 x^{5}}$

Rewrite $8 x^{5}$ as ${(2 x)}^{3} x^{2}$ .

Rewrite $8$ as $2^{3}$ .

$4 x^{\frac{5}{3}} + \sqrt[3]{2^{3} x^{5}}$

Factor out $x^{3}$ .

$4 x^{\frac{5}{3}} + \sqrt[3]{2^{3} (x^{3} x^{2})}$

Rewrite $2^{3} x^{3}$ as ${(2 x)}^{3}$ .

$4 x^{\frac{5}{3}} + \sqrt[3]{{(2 x)}^{3} x^{2}}$

Pull terms out from under the radical.

$4 x^{\frac{5}{3}} + 2 x \sqrt[3]{x^{2}}$