Find the modulus of the following complex numbers
`8-6i^(7)`

Answer

To find the modulus of the complex number \( 8 - 6i^7 \), we can follow these steps:

### Step 1: Simplify \( i^7 \)
Recall that the powers of \( i \) (the imaginary unit) cycle every four terms:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

To simplify \( i^7 \), we can find the equivalent power within the first four:
\[ i^7 = i^{4+3} = i^4 \cdot i^3 = 1 \cdot (-i) = -i \]

### Step 2: Substitute \( i^7 \) back into the expression
Now we substitute \( -i \) for \( i^7 \) in the original expression:
\[ 8 - 6i^7 = 8 - 6(-i) = 8 + 6i \]

### Step 3: Find the modulus of the complex number \( 8 + 6i \)
The modulus of a complex number \( a + bi \) is given by the formula:
\[ |a + bi| = \sqrt{a^2 + b^2} \]
In our case, \( a = 8 \) and \( b = 6 \):
\[ |8 + 6i| = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Final Answer
The modulus of the complex number \( 8 - 6i^7 \) is \( 10 \).

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