Arrange the following numbers in increasing order (smallest first, biggest last):
1.
2.
3.
4.
5.
A
=
8
1
/
4
4
1
/
8
B
=
5
√
128
⋅
4
−
1
/
3
C
=
(
2
4
1
/
3
)
2
D
=
3
√
4
−
1
32
−
1
/
2
E
=
√
2
1
/
2
⋅
4
1
/
4
8
1
/
8


To arrange the given numbers in increasing order, we'll first simplify each one. The given numbers are:

A = \( 8^{1/4} 4^{1/8} \)
B = \( 5 \sqrt{128} \cdot 4^{-1/3} \)
C = \( (24^{1/3})^2 \)
D = \( 3\sqrt{4^{-1/32}} - 1/2 \)
E = \( \sqrt{2^{1/2} \cdot 4^{1/4}} \)

Let's simplify each one:

### Simplifying A:
\( A = 8^{1/4} 4^{1/8} \)

- \( 8 = 2^3 \)
- \( 4 = 2^2 \)

Therefore:
\[ 8^{1/4} = (2^3)^{1/4} = 2^{3/4} \]
\[ 4^{1/8} = (2^2)^{1/8} = 2^{2/8} = 2^{1/4} \]

So:
\[ A = 2^{3/4} \cdot 2^{1/4} = 2^{(3/4 + 1/4)} = 2^1 = 2 \]

### Simplifying B:
\( B = 5 \sqrt{128} \cdot 4^{-1/3} \)

- \( 128 = 2^7 \)
- \( \sqrt{128} = (2^7)^{1/2} = 2^{7/2} = 2^{3.5} \)
- \( 4 = 2^2 \)

Therefore:
\[ 4^{-1/3} = (2^2)^{-1/3} = 2^{-2/3} \]

So:
\[ B = 5 \cdot 2^{3.5} \cdot 2^{-2/3} = 5 \cdot 2^{(3.5 - 2/3)} = 5 \cdot 2^{(3.5 - 0.6667)} = 5 \cdot 2^{2.8333} \]

Approximate value:
\[ 2^{2.8333} \approx 7.8 \]
\[ B \approx 5 \cdot 7.8 = 39 \]

### Simplifying C:
\( C = (24^{1/3})^2 \)

- \( 24^{1/3} = \sqrt[3]{24} \)

Therefore:
\[ C = (\sqrt[3]{24})^2 \]

Approximate value:
\[ \sqrt[3]{24} \approx 2.884 \]
\[ C \approx 2.884^2 \approx 8.32 \]

### Simplifying D:
\( D = 3\sqrt{4^{-1/32}} - 1/2 \)

- \( 4 = 2^2 \)

Therefore:
\[ 4^{-1/32} = (2^2)^{-1/32} = 2^{-2/32} = 2^{-1/16} \]
\[ \sqrt{4^{-1/32}} = \sqrt{2^{-1/16}} = (2^{-1/16})^{1/2} = 2^{-1/32} \]

So:
\[ D = 3 \cdot 2^{-1/32} - 1/2 \]

Since \( 2^{-1/32} \) is very close to 1:
\[ 3 \cdot 2^{-1/32} \approx 3 \]
\[ D \approx 3 - 1/2 = 2.5 \]

### Simplifying E:
\( E = \sqrt{2^{1/2} \cdot 4^{1/4}} \)

- \( 4 = 2^2 \)

Therefore:
\[ 4^{1/4} = (2^2)^{1/4} = 2^{1/2} \]

So:
\[ 2^{1/2} \cdot 4^{1/4} = 2^{1/2} \cdot 2^{1/2} = 2 \]

Thus:
\[ E = \sqrt{2} \]

### Summary of values:
- A = 2
- B ≈ 39
- C ≈ 8.32
- D ≈ 2.5
- E = √2 ≈ 1.414

### Arranging in increasing order:
\[ E < A < D < C < B \]

Therefore, the numbers in increasing order are:
\[ \sqrt{2} < 2 < 2.5 < 8.32 < 39 \]

Would you like more details or have any questions about the steps? Here are some related questions to explore:

1. What is the general process for simplifying exponents?
2. How do you approximate cube roots?
3. What are the properties of exponents used in simplification?
4. How do you convert radicals to exponential form?
5. What are the common mistakes to avoid when simplifying complex expressions?
6. How do you compare numbers with different bases and exponents?
7. How do you approximate decimal values of radicals and exponents?
8. What are practical applications of understanding exponents and radicals?

**Tip:** When simplifying expressions with exponents, always try to express all terms with the same base for easier comparison and calculation.

Learn how to arrange numbers involving exponents and radicals with a detailed step-by-step guide. This problem covers key mathematical concepts such as exponents, radicals, and approximation techniques. Suitable for advanced high school students and anyone interested in mathematical problem-solving.

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