Simplify the Radical Expression sixth root of 27x^15y^12z^3

$\sqrt[6]{27 x^{15} y^{12} z^{3}}$

Step 1

Rewrite $27 x^{15} y^{12} z^{3}$ as ${(x^{2} y^{2})}^{6} \cdot (27 x^{3} z^{3})$ .

Factor out $x^{12}$ .

$\sqrt[6]{27 (x^{12} x^{3}) y^{12} z^{3}}$

Rewrite $x^{12}$ as ${(x^{2})}^{6}$ .

$\sqrt[6]{27 ({(x^{2})}^{6} x^{3}) y^{12} z^{3}}$

Rewrite $y^{12}$ as ${(y^{2})}^{6}$ .

$\sqrt[6]{27 ({(x^{2})}^{6} x^{3}) {(y^{2})}^{6} z^{3}}$

Move $x^{3}$ .

$\sqrt[6]{27 ({(x^{2})}^{6}) {(y^{2})}^{6} x^{3} z^{3}}$

Move $27$ .

$\sqrt[6]{({(x^{2})}^{6}) {(y^{2})}^{6} \cdot 27 x^{3} z^{3}}$

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

$\sqrt[6]{{(x^{2} y^{2})}^{6} \cdot 27 x^{3} z^{3}}$

Add parentheses.

$\sqrt[6]{{(x^{2} y^{2})}^{6} \cdot 27 (x^{3} z^{3})}$

Add parentheses.

$\sqrt[6]{{(x^{2} y^{2})}^{6} \cdot (27 x^{3} z^{3})}$

Step 2

Pull terms out from under the radical.

$| x^{2} y^{2} | \sqrt[6]{27 x^{3} z^{3}}$

Step 3

Remove non-negative terms from the absolute value.

$x^{2} y^{2} \sqrt[6]{27 x^{3} z^{3}}$

Step 4

Rewrite $27 x^{3} z^{3}$ as ${(3 x z)}^{3}$ .

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

Step 5

Rewrite $\sqrt[6]{{(3 x z)}^{3}}$ as $\sqrt{\sqrt[3]{{(3 x z)}^{3}}}$ .

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

Step 6

Pull terms out from under the radical.

$x^{2} y^{2} \sqrt{3 x z}$