Simplify 10x^3 square root of 50x+8x square root of 32x^5

$10 x^{3} \sqrt{50 x} + 8 x \sqrt{32 x^{5}}$

Simplify each term.

Rewrite $50 x$ as $5^{2} \cdot (2 x)$ .

Factor $25$ out of $50$ .

$10 x^{3} \sqrt{25 (2) x} + 8 x \sqrt{32 x^{5}}$

Rewrite $25$ as $5^{2}$ .

$10 x^{3} \sqrt{5^{2} \cdot 2 x} + 8 x \sqrt{32 x^{5}}$

Add parentheses.

$10 x^{3} \sqrt{5^{2} \cdot (2 x)} + 8 x \sqrt{32 x^{5}}$

Pull terms out from under the radical.

$10 x^{3} (5 \sqrt{2 x}) + 8 x \sqrt{32 x^{5}}$

Multiply $5$ by $10$ .

$50 x^{3} \sqrt{2 x} + 8 x \sqrt{32 x^{5}}$

Rewrite $32 x^{5}$ as ${(4 x^{2})}^{2} \cdot (2 x)$ .

Factor $16$ out of $32$ .

$50 x^{3} \sqrt{2 x} + 8 x \sqrt{16 (2) x^{5}}$

Rewrite $16$ as $4^{2}$ .

$50 x^{3} \sqrt{2 x} + 8 x \sqrt{4^{2} \cdot 2 x^{5}}$

Factor out $x^{4}$ .

$50 x^{3} \sqrt{2 x} + 8 x \sqrt{4^{2} \cdot 2 (x^{4} x)}$

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

$50 x^{3} \sqrt{2 x} + 8 x \sqrt{4^{2} \cdot 2 ({(x^{2})}^{2} x)}$

Move $2$ .

$50 x^{3} \sqrt{2 x} + 8 x \sqrt{4^{2} {(x^{2})}^{2} \cdot 2 x}$

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

$50 x^{3} \sqrt{2 x} + 8 x \sqrt{{(4 x^{2})}^{2} \cdot 2 x}$

Add parentheses.

$50 x^{3} \sqrt{2 x} + 8 x \sqrt{{(4 x^{2})}^{2} \cdot (2 x)}$

Pull terms out from under the radical.

$50 x^{3} \sqrt{2 x} + 8 x (4 x^{2} \sqrt{2 x})$

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

Move $x^{2}$ .

$50 x^{3} \sqrt{2 x} + 8 (x^{2} x) (4 \sqrt{2 x})$

Multiply $x^{2}$ by $x$ .

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

$50 x^{3} \sqrt{2 x} + 8 (x^{2} x^{1}) (4 \sqrt{2 x})$

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

$50 x^{3} \sqrt{2 x} + 8 x^{2 + 1} (4 \sqrt{2 x})$

Add $2$ and $1$ .

$50 x^{3} \sqrt{2 x} + 8 x^{3} (4 \sqrt{2 x})$

Multiply $4$ by $8$ .

$50 x^{3} \sqrt{2 x} + 32 x^{3} \sqrt{2 x}$

Add $50 x^{3} \sqrt{2 x}$ and $32 x^{3} \sqrt{2 x}$ .

$82 x^{3} \sqrt{2 x}$