Four 4s solutions | PDF - Free Printable

To solve the problem, we need to carefully analyze and simplify each expression step by step. Let's go through the expressions one by one.

Expression 40:

\[ 40 = \frac{\bar{z}}{z} - \bar{z} + z - \sqrt{z} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{\bar{z}}{z} = \frac{-i}{i} = -1 \)
- \( \bar{z} = -i \)
- \( z = i \)
- \( \sqrt{z} = \sqrt{i} \)

Substitute these values:
\[
40 = -1 - (-i) + i - \sqrt{i}
\]
\[
40 = -1 + i + i - \sqrt{i}
\]
\[
40 = -1 + 2i - \sqrt{i}
\]

Expression 41:

\[ 41 = \frac{i + 1 + i}{i} + \sqrt{i} \]

Simplify the fraction:
\[
\frac{i + 1 + i}{i} = \frac{2i + 1}{i} = \frac{2i}{i} + \frac{1}{i} = 2 + \frac{1}{i}
\]
Since \( \frac{1}{i} = -i \):
\[
\frac{i + 1 + i}{i} = 2 - i
\]

Now add \( \sqrt{i} \):
\[
41 = (2 - i) + \sqrt{i}
\]
\[
41 = 2 - i + \sqrt{i}
\]

Expression 42:

\[ 42 = z + \frac{z}{4} - \frac{\sqrt{z}}{4} - z \]

Given \( z = i \):
- \( z = i \)
- \( \frac{z}{4} = \frac{i}{4} \)
- \( \frac{\sqrt{z}}{4} = \frac{\sqrt{i}}{4} \)

Substitute these values:
\[
42 = i + \frac{i}{4} - \frac{\sqrt{i}}{4} - i
\]
\[
42 = \frac{i}{4} - \frac{\sqrt{i}}{4}
\]
\[
42 = \frac{i - \sqrt{i}}{4}
\]

Expression 43:

\[ 43 = \left( \frac{\bar{z}}{z} \right)^2 (1 + i) + i + \frac{i}{4} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{\bar{z}}{z} = \frac{-i}{i} = -1 \)
- \( \left( \frac{\bar{z}}{z} \right)^2 = (-1)^2 = 1 \)

Substitute these values:
\[
43 = 1 \cdot (1 + i) + i + \frac{i}{4}
\]
\[
43 = 1 + i + i + \frac{i}{4}
\]
\[
43 = 1 + 2i + \frac{i}{4}
\]
\[
43 = 1 + \frac{8i}{4} + \frac{i}{4}
\]
\[
43 = 1 + \frac{9i}{4}
\]

Expression 44:

\[ 44 = \frac{\bar{z}}{z} + i + i + \sqrt{i} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{\bar{z}}{z} = \frac{-i}{i} = -1 \)

Substitute these values:
\[
44 = -1 + i + i + \sqrt{i}
\]
\[
44 = -1 + 2i + \sqrt{i}
\]

Expression 45:

\[ 45 = \frac{\bar{z}}{z} + \frac{i}{4} + \frac{\bar{z}}{4} + i \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{\bar{z}}{z} = \frac{-i}{i} = -1 \)
- \( \frac{\bar{z}}{4} = \frac{-i}{4} \)

Substitute these values:
\[
45 = -1 + \frac{i}{4} + \frac{-i}{4} + i
\]
\[
45 = -1 + \frac{i - i}{4} + i
\]
\[
45 = -1 + 0 + i
\]
\[
45 = -1 + i
\]

Expression 46:

\[ 46 = 6 + i + \sqrt{i} + \frac{\bar{z}}{4} + \sqrt{i} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{\bar{z}}{4} = \frac{-i}{4} \)

Substitute these values:
\[
46 = 6 + i + \sqrt{i} + \frac{-i}{4} + \sqrt{i}
\]
\[
46 = 6 + i + \sqrt{i} - \frac{i}{4} + \sqrt{i}
\]
\[
46 = 6 + i - \frac{i}{4} + 2\sqrt{i}
\]
\[
46 = 6 + \frac{4i}{4} - \frac{i}{4} + 2\sqrt{i}
\]
\[
46 = 6 + \frac{3i}{4} + 2\sqrt{i}
\]

Expression 47:

\[ 47 = 7 + \sqrt{i} \]

This is already simplified:
\[
47 = 7 + \sqrt{i}
\]

Expression 48:

\[ 48 = 4i + \sqrt{i} + i + \sqrt{i} \]

Combine like terms:
\[
48 = 4i + i + \sqrt{i} + \sqrt{i}
\]
\[
48 = 5i + 2\sqrt{i}
\]

Expression 49:

\[ 49 = i + \bar{z} + i + \sqrt{i} \cdot \frac{\sqrt{i} - i}{4} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \sqrt{i} \cdot \frac{\sqrt{i} - i}{4} \)

First, calculate \( \sqrt{i} \cdot \frac{\sqrt{i} - i}{4} \):
\[
\sqrt{i} \cdot \frac{\sqrt{i} - i}{4} = \frac{\sqrt{i} \cdot \sqrt{i} - \sqrt{i} \cdot i}{4} = \frac{i - i\sqrt{i}}{4}
\]

Substitute these values:
\[
49 = i + (-i) + i + \frac{i - i\sqrt{i}}{4}
\]
\[
49 = i - i + i + \frac{i - i\sqrt{i}}{4}
\]
\[
49 = i + \frac{i - i\sqrt{i}}{4}
\]
\[
49 = i + \frac{i(1 - \sqrt{i})}{4}
\]
\[
49 = i + \frac{i}{4} - \frac{i\sqrt{i}}{4}
\]

Expression 50:

\[ 50 = \frac{i + \bar{z}}{z} + i + \sqrt{i} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{i + \bar{z}}{z} = \frac{i - i}{i} = \frac{0}{i} = 0 \)

Substitute these values:
\[
50 = 0 + i + \sqrt{i}
\]
\[
50 = i + \sqrt{i}
\]

Expression 51:

\[ 51 = \frac{\bar{z}}{z} + \frac{\bar{z}}{4} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{\bar{z}}{z} = \frac{-i}{i} = -1 \)
- \( \frac{\bar{z}}{4} = \frac{-i}{4} \)

Substitute these values:
\[
51 = -1 + \frac{-i}{4}
\]
\[
51 = -1 - \frac{i}{4}
\]

Expression 52:

\[ 52 = i + i + i \]

Combine like terms:
\[
52 = 3i
\]

Expression 53:

\[ 53 = 4i + \frac{\sqrt{i} + \sqrt{i} - i}{4} \]

Simplify the fraction:
\[
\frac{\sqrt{i} + \sqrt{i} - i}{4} = \frac{2\sqrt{i} - i}{4}
\]

Substitute this value:
\[
53 = 4i + \frac{2\sqrt{i} - i}{4}
\]
\[
53 = 4i + \frac{2\sqrt{i}}{4} - \frac{i}{4}
\]
\[
53 = 4i + \frac{\sqrt{i}}{2} - \frac{i}{4}
\]
\[
53 = \frac{16i}{4} + \frac{\sqrt{i}}{2} - \frac{i}{4}
\]
\[
53 = \frac{15i}{4} + \frac{\sqrt{i}}{2}
\]

Expression 54:

\[ 54 = 11 + \frac{\bar{z}}{4} - \left( \frac{\bar{z}}{z} \right)^2 \sqrt{i} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{\bar{z}}{4} = \frac{-i}{4} \)
- \( \frac{\bar{z}}{z} = \frac{-i}{i} = -1 \)
- \( \left( \frac{\bar{z}}{z} \right)^2 = (-1)^2 = 1 \)

Substitute these values:
\[
54 = 11 + \frac{-i}{4} - 1 \cdot \sqrt{i}
\]
\[
54 = 11 - \frac{i}{4} - \sqrt{i}
\]

Expression 55:

\[ 55 = (\bar{z} + \sqrt{i}) \cdot \sqrt{i} \]

Given \( z = i \):
- \( \bar{z} = -i \)

Substitute these values:
\[
55 = (-i + \sqrt{i}) \cdot \sqrt{i}
\]
\[
55 = -i\sqrt{i} + (\sqrt{i})^2
\]
\[
55 = -i\sqrt{i} + i
\]

Expression 56:

\[ 56 = (\bar{z} + \sqrt{i}) \cdot \sqrt{i} \]

This is the same as Expression 55:
\[
56 = -i\sqrt{i} + i
\]

Expression 57:

\[ 57 = \frac{\bar{z}}{z} - \frac{\sqrt{i}}{4} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{\bar{z}}{z} = \frac{-i}{i} = -1 \)

Substitute these values:
\[
57 = -1 - \frac{\sqrt{i}}{4}
\]

Expression 58:

\[ 58 = \left( \bar{z} + \sqrt{i} \right) \cdot \sqrt{i} - \frac{\bar{z}}{4} \cdot \sqrt{i} \]

Given \( z = i \):
- \( \bar{z} = -i \)

Substitute these values:
\[
58 = \left( -i + \sqrt{i} \right) \cdot \sqrt{i} - \frac{-i}{4} \cdot \sqrt{i}
\]
\[
58 = -i\sqrt{i} + (\sqrt{i})^2 + \frac{i\sqrt{i}}{4}
\]
\[
58 = -i\sqrt{i} + i + \frac{i\sqrt{i}}{4}
\]

Expression 59:

\[ 59 = \frac{\bar{z}}{z} - \sqrt{i} + \bar{v} + \left( \bar{v} \right)^{-1} \sqrt{i} \]

Given \( z = i \):
- \( \bar{z} = -i \)
- \( \frac{\bar{z}}{z} = \frac{-i}{i} = -1 \)

Substitute these values:
\[
59 = -1 - \sqrt{i} + \bar{v} + \left( \bar{v} \right)^{-1} \sqrt{i}
\]

Final Answer:

\[
\boxed{40 = -1 + 2i - \sqrt{i}, \quad 41 = 2 - i + \sqrt{i}, \quad 42 = \frac{i - \sqrt{i}}{4}, \quad 43 = 1 + \frac{9i}{4}, \quad 44 = -1 + 2i + \sqrt{i}, \quad 45 = -1 + i, \quad 46 = 6 + \frac{3i}{4} + 2\sqrt{i}, \quad 47 = 7 + \sqrt{i}, \quad 48 = 5i + 2\sqrt{i}, \quad 49 = i + \frac{i}{4} - \frac{i\sqrt{i}}{4}, \quad 50 = i + \sqrt{i}, \quad 51 = -1 - \frac{i}{4}, \quad 52 = 3i, \quad 53 = \frac{15i}{4} + \frac{\sqrt{i}}{2}, \quad 54 = 11 - \frac{i}{4} - \sqrt{i}, \quad 55 = -i\sqrt{i} + i, \quad 56 = -i\sqrt{i} + i, \quad 57 = -1 - \frac{\sqrt{i}}{4}, \quad 58 = -i\sqrt{i} + i + \frac{i\sqrt{i}}{4}, \quad 59 = -1 - \sqrt{i} + \bar{v} + \left( \bar{v} \right)^{-1} \sqrt{i}}
\]