Question

2
Find the value of (3+2/i)(i⁶-i⁷)(1+i¹¹)
​

Answer 1

Answer:

6i+4

Step-by-step explanation:

The given expression is (3+\frac{2}{i}))i^6-i^7)(1+i^{11})(3+

i

2

))i

6

−i

7

)(1+i

11

)

We can rewrite this expression as

(3+\frac{2}{i})((i^2)^3-(i^2)^3i(1+(i^2)^5i)(3+

i

2

)((i

2

)

3

−(i

2

)

3

i(1+(i

2

)

5

i)

We know that i^2=-1i

2

=−1

Thus, the expression becomes

\begin{gathered}(3+\frac{2}{i})((-1)^3-(-1)^3i(1+(-1)^5i)\\\\=(3+\frac{2}{i})(-1+i)(1-i)\end{gathered}

(3+

i

2

)((−1)

3

−(−1)

3

i(1+(−1)

5

i)

=(3+

i

2

)(−1+i)(1−i)

Now. rationalize the denominator by multiplying numerator and denominator by i

\begin{gathered}(3+\frac{2}{i}\cdot\frac{i}{i})(-1+i)(1-i)\\\\=(3+\frac{2i}{i^2}(-1+i)(1-i)\\\\=(2i-3)(1-i)(1-i)\\\\=(2i-3)(1+i^2-2i)\\\\=(2i-3)(1-1-2i)\\\\=2i(3-2i)\\\\=6i-4i^2\\\\=6i+4\\\\\end{gathered}

(3+

i

2

⋅

i

i

)(−1+i)(1−i)

=(3+

i

2

2i

(−1+i)(1−i)

=(2i−3)(1−i)(1−i)

=(2i−3)(1+i

2

−2i)

=(2i−3)(1−1−2i)

=2i(3−2i)

=6i−4i

2

=6i+4

Therefore, the simplified form is 6i+4