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Let's work through each part of the question to determine the value of \( y \) for each case:

---

### (a) \( y = \sqrt[4]{192} \)
This is asking for the 4th root of 192. Let's simplify it:

1. Factor 192: 
   \[
   192 = 2^6 \times 3
   \]

2. Apply the 4th root:
   \[
   \sqrt[4]{192} = \sqrt[4]{2^6 \times 3} = \sqrt[4]{2^6} \times \sqrt[4]{3}
   \]
   \[
   \sqrt[4]{2^6} = 2^{6/4} = 2^{3/2} = 2 \times \sqrt{2}
   \]
   So:
   \[
   \sqrt[4]{192} = 2\sqrt{2} \times \sqrt[4]{3}
   \]

This simplifies to \( y = 2\sqrt{2} \times \sqrt[4]{3} \), but this is the simplest form unless a numerical value is needed.

---

### (b) \( y = \sqrt[3]{32^9} \)
This is asking for the cube root of \( 32^9 \):

1. Express 32 as a power of 2:
   \[
   32 = 2^5
   \]

2. Now the equation becomes:
   \[
   y = \sqrt[3]{(2^5)^9} = \sqrt[3]{2^{45}}
   \]

3. Apply the cube root:
   \[
   y = 2^{45/3} = 2^{15}
   \]

Thus, \( y = 2^{15} = 32768 \).

---

### (c) \( y = \sqrt[4]{16^8} \)
This is asking for the 4th root of \( 16^8 \):

1. Express 16 as a power of 2:
   \[
   16 = 2^4
   \]

2. Now the equation becomes:
   \[
   y = \sqrt[4]{(2^4)^8} = \sqrt[4]{2^{32}}
   \]

3. Apply the 4th root:
   \[
   y = 2^{32/4} = 2^8
   \]

Thus, \( y = 2^8 = 256 \).

---

### (d) \( y = \sqrt[3]{27^4} \)
This is asking for the cube root of \( 27^4 \):

1. Express 27 as a power of 3:
   \[
   27 = 3^3
   \]

2. Now the equation becomes:
   \[
   y = \sqrt[3]{(3^3)^4} = \sqrt[3]{3^{12}}
   \]

3. Apply the cube root:
   \[
   y = 3^{12/3} = 3^4
   \]

Thus, \( y = 3^4 = 81 \).

---

### Summary of the values:
- (a) \( y = 2\sqrt{2} \times \sqrt[4]{3} \)
- (b) \( y = 32768 \)
- (c) \( y = 256 \)
- (d) \( y = 81 \)

Let me know if you'd like more details or have any questions!

Here are some related questions to expand on this:
1. What is the general formula for simplifying roots with exponents?
2. How can we estimate roots like \( \sqrt[4]{192} \) using a calculator?
3. What are the applications of cube roots in real-life situations?
4. How do fractional exponents relate to radical expressions?
5. What methods can we use to factor large numbers like 192 efficiently?

**Tip:** To simplify expressions with roots, always try to factor the base into prime numbers, then apply the root laws!

Tentukan nilai y dari:
(a) y = 4√192
(b) y = ³√32⁹
(c) y = ⁴√16⁸
(d) y = ³√27⁴

Learn how to simplify expressions with roots and powers, including 4th and 3rd roots. This problem focuses on breaking down expressions like ³√32⁹ and ⁴√16⁸. Suitable for high school math students.

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