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The given values are \( 27^{\frac{2}{3}} \), \( 64^{\frac{1}{3}} \), and \( 4^{\frac{3}{2}} \). Let's first compute each of these values and then arrange them in ascending order.

### 1. \( 27^{\frac{2}{3}} \)
We can break this down:
\[
27 = 3^3, \quad \text{so} \quad 27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^2 = 9
\]

### 2. \( 64^{\frac{1}{3}} \)
We know that:
\[
64 = 4^3, \quad \text{so} \quad 64^{\frac{1}{3}} = (4^3)^{\frac{1}{3}} = 4
\]

### 3. \( 4^{\frac{3}{2}} \)
We can express this as:
\[
4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}} = 2^3 = 8
\]

Now, the values are:
- \( 27^{\frac{2}{3}} = 9 \)
- \( 64^{\frac{1}{3}} = 4 \)
- \( 4^{\frac{3}{2}} = 8 \)

### Ascending Order
Arranging them from smallest to largest:
\[
64^{\frac{1}{3}} = 4, \quad 4^{\frac{3}{2}} = 8, \quad 27^{\frac{2}{3}} = 9
\]

Thus, the ascending order is:
\[
64^{\frac{1}{3}}, \quad 4^{\frac{3}{2}}, \quad 27^{\frac{2}{3}}
\]

Would you like further details or have any questions?

### Related Questions:
1. How would you solve a general expression like \( a^{m/n} \)?
2. What is the general method to compare fractional exponents?
3. What is the relationship between roots and fractional powers?
4. How does logarithmic comparison work for similar problems?
5. Can fractional exponents ever produce negative results?

### Tip:
When simplifying expressions with fractional exponents, always try to express the base in terms of prime factors to make calculations easier.

Put the following values into ascending order: 27^(2/3), 64^(1/3), and 4^(3/2).

Learn how to arrange 27^(2/3), 64^(1/3), and 4^(3/2) in ascending order. This solution demonstrates key algebraic concepts involving exponents and roots. Suitable for students in Grades 8-10.

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