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

Let's evaluate each expression to determine the numerical values and then arrange them in increasing order.

### Expression A:
\[ A = 8 \cdot \frac{1}{4} + 4 \cdot \frac{1}{8} \]
This simplifies to:
\[ A = 2 + 0.5 = 2.5 \]

### Expression B:
\[ B = 5 \sqrt{128} \cdot 4 - \frac{1}{3} \]
Let's break this down:
- \( \sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2} \)
- Therefore, \( 5 \sqrt{128} = 40\sqrt{2} \)
- \( B = 40\sqrt{2} \cdot 4 - \frac{1}{3} = 160\sqrt{2} - \frac{1}{3} \)

### Expression C:
\[ C = \left( 2 \cdot \frac{4}{3} \right)^2 \]
This simplifies to:
\[ C = \left(\frac{8}{3}\right)^2 = \frac{64}{9} \approx 7.11 \]

### Expression D:
\[ D = 3 \sqrt{4} - \frac{1}{32} - \frac{1}{2} \]
Let's break this down:
- \( \sqrt{4} = 2 \)
- Therefore, \( D = 6 - \frac{1}{32} - 0.5 = 5.5 - \frac{1}{32} \approx 5.46875 \)

### Expression E:
\[ E = \sqrt{2 \cdot \frac{1}{2}} \cdot 4 \cdot \frac{1}{4} \cdot 8 \cdot \frac{1}{8} \]
This simplifies to:
- \( E = \sqrt{1} \cdot 1 \cdot 2 \approx 2 \)

### Arranging the Numbers:

To arrange these values:
- \( A = 2.5 \)
- \( B = 160\sqrt{2} - \frac{1}{3} \approx 224.407 - 0.333 = 224.074 \)
- \( C \approx 7.11 \)
- \( D \approx 5.46875 \)
- \( E = 2 \)

In increasing order:

\[ E < A < D < C < B \]

So the order is:

\[ E, A, D, C, B \]

Would you like a detailed explanation of any of these calculations? Here are some related questions to explore:

1. How to calculate square roots of large numbers?
2. What is the process to simplify complex mathematical expressions?
3. What are some methods to approximate square roots?
4. How do you solve equations involving multiple operations?
5. How does one approach comparing irrational numbers?
6. What are common pitfalls when simplifying expressions with roots?
7. How to use approximation for quick calculations in exams?
8. How does one convert between different forms of fractions and roots?

**Tip:** When arranging numbers, simplifying and approximating them can make comparisons easier.

Learn how to arrange numbers in increasing order using complex mathematical expressions. This problem involves arithmetic operations, square roots, and approximation techniques. Detailed step-by-step solution provided.

Math Problem Statement

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

Solution

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