Question

which expression is equivalent to \\(\sqrt{128x^8y^3z^9}\\)? assume \\(y \geq 0\\) and \\(z \geq 0\\).
options:
\\(8x^4yz^4\sqrt{2yz}\\)
\\(4x^2yz^3\sqrt{2x^2}\\)
\\(2x^2z^2\sqrt{8y^3z}\\)
\\(64x^4yz^4\sqrt{2yz}\\)

Explanation:

Step1: Factor the radicand

Factor \(128x^8y^9z^9\) into perfect squares and remaining factors:
\(128 = 64\times2\), \(x^8=(x^4)^2\), \(y^9 = y^8\times y=(y^4)^2\times y\), \(z^9=z^8\times z=(z^4)^2\times z\).
So, \(\sqrt{128x^8y^9z^9}=\sqrt{64\times2\times(x^4)^2\times(y^4)^2\times y\times(z^4)^2\times z}\).

Step2: Simplify the square root

Use \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a,b\geq0\)):
\(\sqrt{64}\cdot\sqrt{(x^4)^2}\cdot\sqrt{(y^4)^2}\cdot\sqrt{(z^4)^2}\cdot\sqrt{2yz}\).
Simplify each perfect square:
\(8\cdot x^4\cdot y^4\cdot z^4\cdot\sqrt{2yz}\).
Multiply the coefficients and variables outside:
\(8x^4y^4z^4\sqrt{2yz}\).

Answer:

\(8x^4yz^4\sqrt{2yz}\) (matching the top - middle option)