Answer: The correct option is (C) [tex]8x^4yz^4\sqrt{2yz}.[/tex]
Step-by-step explanation: We are given to select the correct expression that is equivalent to the expression below:
[tex]E=\sqrt{128x^8y^3z^9}.[/tex]
We will be using the following properties of exponents:
[tex](i)~\sqrt a=a^\frac{1}{2},\\\\(ii)~(ab)^x=a^xb^x,\\\\(iii)~(a^x)^y=a^{x\times y}.[/tex]
Since y > 0 and z > 0, so we will be considering the positive square root of y and z.
We have
[tex]E\\\\=\sqrt{128x^8y^3z^9}\\\\=(128x^8y^3z^9)^\frac{1}{2}\\\\=(128)^\frac{1}{2}(x^8)^\frac{1}{2}(y^3)^\frac{1}{2}(z^9)^\frac{1}{2}\\\\=(2\times 8^2)^\frac{1}{2}x^{8\times \frac{1}{2}}y^{3\times\frac{1}{2}}z^{9\times\frac{1}{2}}\\\\=8x^4yz^4(2yz)^\frac{1}{2}\\\\=8x^4yz^4\sqrt{2yz}.[/tex]
Thus, the equivalent expression is [tex]8x^4yz^4\sqrt{2yz}.[/tex]
(C) is the correct option.
