Simplify 8x^3y square root of 9x^2y^6

$8 x^{3} y \sqrt{9 x^{2} y^{6}}$

Step 1

Rewrite $9 x^{2} y^{6}$ as ${(3 x y^{3})}^{2}$ .

$8 x^{3} y \sqrt{{(3 x y^{3})}^{2}}$

Step 2

Pull terms out from under the radical, assuming positive real numbers.

$8 x^{3} y (3 x y^{3})$

Step 3

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

Move $x$ .

$8 (x \cdot x^{3}) y (3 y^{3})$

Multiply $x$ by $x^{3}$ .

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

$8 (x^{1} x^{3}) y (3 y^{3})$

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

$8 x^{1 + 3} y (3 y^{3})$

Add $1$ and $3$ .

$8 x^{4} y (3 y^{3})$

Step 4

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

Move $y^{3}$ .

$8 x^{4} (y^{3} y) \cdot 3$

Multiply $y^{3}$ by $y$ .

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

$8 x^{4} (y^{3} y^{1}) \cdot 3$

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

$8 x^{4} y^{3 + 1} \cdot 3$

Add $3$ and $1$ .

$8 x^{4} y^{4} \cdot 3$

Step 5

Multiply $3$ by $8$ .

$24 x^{4} y^{4}$