 Simplify (1 + 2 /𝑖 ) (1 + 3 /𝑖 ) (2 + 𝑖) ^−3 and express it in the form of a +ib.

To simplify the expression \((1 + \frac{2}{i})(1 + \frac{3}{i})(2 + i)^{-3}\) and express it in the form \(a + ib\), let's break it down step-by-step.

### Step 1: Simplify each part
First, let's deal with the terms containing \(i\).

#### a) Simplifying \(1 + \frac{2}{i}\)
We can multiply the numerator and denominator by \(i\) to remove the \(i\) from the denominator:
\[
1 + \frac{2}{i} = 1 - 2i
\]
(since \(\frac{1}{i} = -i\)).

#### b) Simplifying \(1 + \frac{3}{i}\)
Similarly, we multiply the numerator and denominator by \(i\):
\[
1 + \frac{3}{i} = 1 - 3i
\]

So, the expression becomes:
\[
(1 - 2i)(1 - 3i)(2 + i)^{-3}
\]

### Step 2: Simplify the product \((1 - 2i)(1 - 3i)\)
To multiply two complex numbers, we apply the distributive property:
\[
(1 - 2i)(1 - 3i) = 1(1) + 1(-3i) + (-2i)(1) + (-2i)(-3i)
\]
\[
= 1 - 3i - 2i + 6i^2
\]
Since \(i^2 = -1\), this simplifies further:
\[
= 1 - 3i - 2i + 6(-1)
= 1 - 3i - 2i - 6
= -5 - 5i
\]

### Step 3: Simplify \((2 + i)^{-3}\)
To find \((2 + i)^{-3}\), first we need to find \((2 + i)^3\).

#### a) Calculate \((2 + i)^2\)
\[
(2 + i)^2 = (2 + i)(2 + i) = 2(2) + 2(i) + i(2) + i(i) = 4 + 2i + 2i + i^2
\]
\[
= 4 + 4i + (-1) = 3 + 4i
\]

#### b) Now calculate \((2 + i)^3\)
\[
(2 + i)^3 = (2 + i)(3 + 4i) = 2(3) + 2(4i) + i(3) + i(4i)
\]
\[
= 6 + 8i + 3i + 4i^2 = 6 + 11i + 4(-1) = 6 + 11i - 4 = 2 + 11i
\]

Thus, \((2 + i)^3 = 2 + 11i\).

#### c) Find \((2 + i)^{-3}\)
The inverse of a complex number \(a + bi\) is given by:
\[
\frac{1}{a + bi} = \frac{a - bi}{a^2 + b^2}
\]
Here, \(a = 2\) and \(b = 11\), so:
\[
(2 + 11i)^{-1} = \frac{2 - 11i}{2^2 + 11^2} = \frac{2 - 11i}{4 + 121} = \frac{2 - 11i}{125}
\]
\[
= \frac{2}{125} - \frac{11i}{125}
\]
\[
= 0.016 - 0.088i
\]

Thus, \((2 + i)^{-3} = 0.016 - 0.088i\).

### Step 4: Multiply the results
Now, we multiply \((-5 - 5i)\) by \((0.016 - 0.088i)\):
\[
(-5 - 5i)(0.016 - 0.088i) = (-5)(0.016) + (-5)(-0.088i) + (-5i)(0.016) + (-5i)(-0.088i)
\]
\[
= -0.08 + 0.44i - 0.08i + 0.44i^2
\]
Since \(i^2 = -1\), this becomes:
\[
-0.08 + 0.44i - 0.08i + 0.44(-1) = -0.08 + 0.44i - 0.08i - 0.44
\]
\[
= -0.52 + 0.36i
\]

### Final Answer:
The simplified form of \((1 + \frac{2}{i})(1 + \frac{3}{i})(2 + i)^{-3}\) is:
\[
-0.52 + 0.36i
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you simplify expressions involving complex numbers?
2. What is the geometric interpretation of complex numbers?
3. How do you find the inverse of a complex number?
4. Can you explain the modulus and argument of a complex number?
5. What are polar coordinates in the context of complex numbers?

**Tip:** Always use \(i^2 = -1\) to simplify powers of \(i\) in complex number calculations!

Simplify (1 + 2/i)(1 + 3/i)(2 + i)^−3 and express it in the form of a + ib.

Step-by-step solution to simplify the complex expression (1 + 2/i)(1 + 3/i)(2 + i)^{-3} and express it in the form of a + ib. Learn key concepts like complex number manipulation, inverse, and powers of i. Suitable for Grades 11-12.

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