Simplify square root of 9x square root of 8x cube root of 8x^3

$\sqrt{9 x \sqrt{8 x \sqrt[3]{8 x^{3}}}}$

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

$\sqrt{9 x \sqrt{8 x \sqrt[3]{{(2 x)}^{3}}}}$

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

$\sqrt{9 x \sqrt{8 x \cdot 2 x}}$

Combine exponents.

Multiply $2$ by $8$ .

$\sqrt{9 x \sqrt{16 x \cdot x}}$

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

$\sqrt{9 x \sqrt{16 (x^{1} x)}}$

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

$\sqrt{9 x \sqrt{16 (x^{1} x^{1})}}$

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

$\sqrt{9 x \sqrt{16 x^{1 + 1}}}$

Add $1$ and $1$ .

$\sqrt{9 x \sqrt{16 x^{2}}}$

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$\sqrt{9 x \sqrt{{(4 x)}^{2}}}$

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

$\sqrt{9 x \cdot 4 x}$

Combine exponents.

Multiply $4$ by $9$ .

$\sqrt{36 x \cdot x}$

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

$\sqrt{36 (x^{1} x)}$

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

$\sqrt{36 (x^{1} x^{1})}$

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

$\sqrt{36 x^{1 + 1}}$

Add $1$ and $1$ .

$\sqrt{36 x^{2}}$

Rewrite $36 x^{2}$ as ${(6 x)}^{2}$ .

$\sqrt{{(6 x)}^{2}}$

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

$6 x$