Simplify 7x^3 cube root of 375x^8y^10

$7 x^{3} \sqrt[3]{375 x^{8} y^{10}}$

Step 1

Rewrite $375 x^{8} y^{10}$ as ${(5 x^{2} y^{3})}^{3} \cdot (3 x^{2} y)$ .

Factor $125$ out of $375$ .

$7 x^{3} \sqrt[3]{125 (3) x^{8} y^{10}}$

Rewrite $125$ as $5^{3}$ .

$7 x^{3} \sqrt[3]{5^{3} \cdot 3 x^{8} y^{10}}$

Factor out $x^{6}$ .

$7 x^{3} \sqrt[3]{5^{3} \cdot 3 (x^{6} x^{2}) y^{10}}$

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$7 x^{3} \sqrt[3]{5^{3} \cdot 3 ({(x^{2})}^{3} x^{2}) y^{10}}$

Factor out $y^{9}$ .

$7 x^{3} \sqrt[3]{5^{3} \cdot 3 ({(x^{2})}^{3} x^{2}) (y^{9} y)}$

Rewrite $y^{9}$ as ${(y^{3})}^{3}$ .

$7 x^{3} \sqrt[3]{5^{3} \cdot 3 ({(x^{2})}^{3} x^{2}) ({(y^{3})}^{3} y)}$

Move $x^{2}$ .

$7 x^{3} \sqrt[3]{5^{3} \cdot 3 ({(x^{2})}^{3}) {(y^{3})}^{3} x^{2} y}$

Move $3$ .

$7 x^{3} \sqrt[3]{(5^{3} ({(x^{2})}^{3})) {(y^{3})}^{3} \cdot 3 x^{2} y}$

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

$7 x^{3} \sqrt[3]{{(5 x^{2} y^{3})}^{3} \cdot 3 x^{2} y}$

Add parentheses.

$7 x^{3} \sqrt[3]{{(5 x^{2} y^{3})}^{3} \cdot 3 (x^{2} y)}$

Add parentheses.

$7 x^{3} \sqrt[3]{{(5 x^{2} y^{3})}^{3} \cdot (3 x^{2} y)}$

Step 2

Pull terms out from under the radical.

$7 x^{3} (5 x^{2} y^{3} \sqrt[3]{3 x^{2} y})$

Step 3

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

Move $x^{2}$ .

$7 (x^{2} x^{3}) (5 y^{3} \sqrt[3]{3 x^{2} y})$

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

$7 x^{2 + 3} (5 y^{3} \sqrt[3]{3 x^{2} y})$

Add $2$ and $3$ .

$7 x^{5} (5 y^{3} \sqrt[3]{3 x^{2} y})$

Step 4

Multiply $5$ by $7$ .

$35 x^{5} (y^{3} \sqrt[3]{3 x^{2} y})$

Step 5

Reorder factors in $35 x^{5} (y^{3} \sqrt[3]{3 x^{2} y})$ .

$35 x^{5} y^{3} \sqrt[3]{3 x^{2} y}$