Let's evaluate each expression using the given values for x, y, and z.

1. \( 4x^{-0.5} - 3y^{1/3} + 2z^{2/3} \)
\( = 4(2)^{-0.5} - 3(3)^{1/3} + 2(4)^{2/3} \)
\( = 4(1/\sqrt{2}) - 3(\sqrt[3]{3}) + 2(\sqrt[3]{4^2}) \)
\( = 4/\sqrt{2} - 3\sqrt[3]{3} + 2\sqrt[3]{16} \)
\( = 2\sqrt{2} - 3\sqrt[3]{3} + 2\sqrt[3]{16} \)

2. \( 6x^{-1} - 2y^{3/2} + 3z^{-2} \)
\( = 6(2)^{-1} - 2(3)^{3/2} + 3(4)^{-2} \)
\( = 6/2 - 2\sqrt{3^3} + 3/4^2 \)
\( = 3 - 2\sqrt{27} + 3/16 \)
\( = 3 - 2(3\sqrt{3}) + 3/16 \)
\( = 3 - 6\sqrt{3} + 3/16 \)

3. \( 2x^{3/4} - y^{-1/2} * z^{4/5} \)
\( = 2(2)^{3/4} - (3)^{-1/2} * (4)^{4/5} \)
\( = 2\sqrt[4]{2^3} - 1/\sqrt{3} * \sqrt[5]{4^4} \)
\( = 2\sqrt[4]{8} - 1/\sqrt{3} * \sqrt[5]{256} \)
\( = 2\sqrt[4]{8} - 1/\sqrt{3} * 4 \)

4. \( x^{1/2} * y^{-4/3} - z^{-1/3} \)
\( = (2)^{1/2} * (3)^{-4/3} - (4)^{-1/3} \)
\( = \sqrt{2} * 1/\sqrt[3]{3^4} - 1/\sqrt[3]{4} \)
\( = \sqrt{2} * 1/\sqrt[3]{81} - 1/\sqrt[3]{4} \)
\( = \sqrt{2} * 1/3\sqrt[3]{3} - 1/\sqrt[3]{4} \)
\( = \sqrt{2}/3\sqrt[3]{3} - 1/\sqrt[3]{4} \)

Now, let's calculate the numerical values for each expression:

1. \( 2\sqrt{2} - 3\sqrt[3]{3} + 2\sqrt[3]{16} \)
\( \approx 2(1.414) - 3(1.442) + 2(2.519) \)
\( \approx 2.828 - 4.326 + 5.038 \)
\( \approx 3.540 \)

2. \( 3 - 6\sqrt{3} + 3/16 \)
\( \approx 3 - 6(1.732) + 0.1875 \)
\( \approx 3 - 10.392 + 0.1875 \)
\( \approx -7.2045 \)

3. \( 2\sqrt[4]{8} - 1/\sqrt{3} * 4 \)
\( \approx 2(1.681) - 1/(1.732) * 4 \)
\( \approx 3.362 - 2.309 \)
\( \approx 1.053 \)

4. \( \sqrt{2}/3\sqrt[3]{3} - 1/\sqrt[3]{4} \)
\( \approx (1.414)/(3 * 1.442) - 1/(1.587) \)
\( \approx 0.327 - 0.630 \)
\( \approx -0.303 \)

The expression with the greatest value is the first one, with an approximate value of 3.540.
주어진 x, y, z 값을 사용하여 각 식을 평가해 보겠습니다.

1. \( 4x^{-0.5} - 3y^{1/3} + 2z^{2/3} \)
\( = 4(2)^{-0.5} - 3(3)^{1/3} + 2(4)^{2/3} \)
\( = 4(1/\sqrt{2}) - 3(\sqrt[3]{3}) + 2(\sqrt[3]{4^2}) \)
\( = 4/\sqrt{2} - 3\sqrt[3]{3} + 2\sqrt[3]{16} \)
\( = 2\sqrt{2} - 3\sqrt[3]{3} + 2\sqrt[3]{16} \)

2. \( 6x^{-1} - 2y^{3/2} + 3z^{-2} \)
\( = 6(2)^{-1} - 2(3)^{3/2} + 3(4)^{-2} \)
\( = 6/2 - 2\sqrt{3^3} + 3/4^2 \)
\( = 3 - 2\sqrt{27} + 3/16 \)
\( = 3 - 2(3\sqrt{3}) + 3/16 \)
\( = 3 - 6\sqrt{3} + 3/16 \)

3. \( 2x^{3/4} - y^{-1/2} * z^{4/5} \)
\( = 2(2)^{3/4} - (3)^{-1/2} * (4)^{4/5} \)
\( = 2\sqrt[4]{2^3} - 1/\sqrt{3} * \sqrt[5]{4^4} \)
\( = 2\sqrt[4]{8} - 1/\sqrt{3} * \sqrt[5]{256} \)
\( = 2\sqrt[4]{8} - 1/\sqrt{3} * 4 \)

4. \( x^{1/2} * y^{-4/3} - z^{-1/3} \)
\( = (2)^{1/2} * (3)^{-4/3} - (4)^{-1/3} \)
\( = \sqrt{2} * 1/\sqrt[3]{3^4} - 1/\sqrt[3]{4} \)
\( = \sqrt{2} * 1/\sqrt[3]{81} - 1/\sqrt[3]{4} \)
\( = \sqrt{2} * 1/3\sqrt[3]{3} - 1/\sqrt[3]{4} \)
\( = \sqrt{2}/3\sqrt[3]{3} - 1/\sqrt[3]{4} \)

이제 각 식의 수치 값을 계산해 보겠습니다:

1. \( 2\sqrt{2} - 3\sqrt[3]{3} + 2\sqrt[3]{16} \)
\( \approx 2(1.414) - 3(1.442) + 2(2.519) \)
\( \approx 2.828 - 4.326 + 5.038 \)
\( \approx 3.540 \)

2. \( 3 - 6\sqrt{3} + 3/16 \)
\( \approx 3 - 6(1.732) + 0.1875 \)
\( \approx 3 - 10.392 + 0.1875 \)
\( \approx -7.2045 \)

3. \( 2\sqrt[4]{8} - 1/\sqrt{3} * 4 \)
\( \approx 2(1.681) - 1/(1.732) * 4 \)
\( \approx 3.362 - 2.309 \)
\( \approx 1.053 \)

4. \( \sqrt{2}/3\sqrt[3]{3} - 1/\sqrt[3]{4} \)
\( \approx (1.414)/(3 * 1.442) - 1/(1.587) \)
\( \approx 0.327 - 0.630 \)
\( \approx -0.303 \)

가장 큰 값을 갖는 식은 3.540의 근사값으로 첫 번째 식입니다.