Simplify 3x square root of 54x^4y^7z^10

$3 x \sqrt{54 x^{4} y^{7} z^{10}}$

Rewrite $54 x^{4} y^{7} z^{10}$ as ${(3 x^{2} y^{3} z^{5})}^{2} \cdot (6 y)$ .

Factor $9$ out of $54$ .

$3 x \sqrt{9 (6) x^{4} y^{7} z^{10}}$

Rewrite $9$ as $3^{2}$ .

$3 x \sqrt{3^{2} \cdot 6 x^{4} y^{7} z^{10}}$

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

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

Factor out $y^{6}$ .

$3 x \sqrt{3^{2} \cdot 6 {(x^{2})}^{2} (y^{6} y) z^{10}}$

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

$3 x \sqrt{3^{2} \cdot 6 {(x^{2})}^{2} ({(y^{3})}^{2} y) z^{10}}$

Rewrite $z^{10}$ as ${(z^{5})}^{2}$ .

$3 x \sqrt{3^{2} \cdot 6 {(x^{2})}^{2} ({(y^{3})}^{2} y) {(z^{5})}^{2}}$

Move $y$ .

$3 x \sqrt{3^{2} \cdot 6 {(x^{2})}^{2} ({(y^{3})}^{2}) {(z^{5})}^{2} y}$

Move $6$ .

$3 x \sqrt{3^{2} {(x^{2})}^{2} ({(y^{3})}^{2}) {(z^{5})}^{2} \cdot 6 y}$

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

$3 x \sqrt{{(3 x^{2} y^{3} z^{5})}^{2} \cdot 6 y}$

Add parentheses.

$3 x \sqrt{{(3 x^{2} y^{3} z^{5})}^{2} \cdot (6 y)}$

Pull terms out from under the radical.

$3 x (3 x^{2} y^{3} z^{5} \sqrt{6 y})$

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

Move $x^{2}$ .

$3 (x^{2} x) (3 y^{3} z^{5} \sqrt{6 y})$

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

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

$3 (x^{2} x^{1}) (3 y^{3} z^{5} \sqrt{6 y})$

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

$3 x^{2 + 1} (3 y^{3} z^{5} \sqrt{6 y})$

Add $2$ and $1$ .

$3 x^{3} (3 y^{3} z^{5} \sqrt{6 y})$

Multiply $3$ by $3$ .

$9 x^{3} y^{3} z^{5} \sqrt{6 y}$