Simplify 7x^3 cube root of 8x^11y^6

$7 x^{3} \sqrt[3]{8 x^{11} y^{6}}$

Step 1

Rewrite $8 x^{11} y^{6}$ as ${(2 x^{3} y^{2})}^{3} x^{2}$ .

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

$7 x^{3} \sqrt[3]{2^{3} x^{11} y^{6}}$

Factor out $x^{9}$ .

$7 x^{3} \sqrt[3]{2^{3} (x^{9} x^{2}) y^{6}}$

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

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

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

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

Move $x^{2}$ .

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

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

Move $x^{3}$ .

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

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

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

Add $3$ and $3$ .

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

Step 4

Multiply $2$ by $7$ .

$14 x^{6} (y^{2} \sqrt[3]{x^{2}})$

Step 5

Reorder factors in $14 x^{6} (y^{2} \sqrt[3]{x^{2}})$ .

$14 x^{6} y^{2} \sqrt[3]{x^{2}}$