Answer

Verified

390.6k+ views

**Hint:**Try to convert the complex number into polar form first. Then apply the DeMoivre’s theorem and write \[{(i - 1)^{10}}\] in the complex form of \[a + bi\], where \[a\,\,and\,\,b\] are the real numbers and these numbers do not use the trigonometric function. After rearranging the term and getting in the general form for the theorem we can proceed with the question.

**Complete step by step solution:**

The given question is \[{( - 1 + i)^{10}}\]. \[{( - 1 + i)^{10}}\] can be also written as \[{(i - 1)^{10}}\]. First write the complex number in polar form, that is in the form of \[a + bi\] and then apply the De-Moivre’s theorem .De-Moivre’s theorem states that:

\[ \Rightarrow x + iy = \sqrt {{x^2} + {y^2}{e^{i\theta }}} \] \[where\,{e^{i\theta }} = cos(\phi ) + i\sin (\phi )\];\[\phi = \arctan \left( {\dfrac{y}{x}} \right)\]

Applying this to our question and after solving, we get:

\[ \Rightarrow - 1 + i = \sqrt {2{e^{ - 1\dfrac{\pi }{4}}}} \]

Now putting the value in our question, proceeding further, on solving we get:

\[ \Rightarrow {(i - 1)^{10}} = {(\sqrt {2{e^{ - i\dfrac{\pi }{4}}}} )^{10}} = {(\surd 2)^{10}}{e^{ - i\dfrac{{10\pi }}{4}}}\]

But,\[{e^{ - i\dfrac{{10\pi }}{4}}} = {e^{ - i\dfrac{{8\pi }}{4}}} \cdot {e^{ - i\dfrac{\pi }{2}}} = {e^{ - i\dfrac{\pi }{4}}}\]

\[ \Rightarrow {(\sqrt 2 )^{10}} = 32\]

\[ \therefore {(i - 1)^{10}} = - 32i\]

So, this is the final answer in the polar form of the given complex number.

**Hence, we finally get \[{(i - 1)^{10}} = - 32i\] as the answer.**

**Additional information:**

Dealing with the complex equation you have to be careful only when you are dealing in a higher degree equation because there the value of “iota” is given as for higher degree terms and accordingly the question needs to be solved and simplification can be done.

**Note:**After the development of iota, research leads with the formulas associated and the properties like summation, subtraction, multiplication and division for the complex numbers. Graphs for complex numbers are also designed and the area under which graph is drawn contains complex numbers only, but the relation between complex and real numbers can be drawn.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

10 examples of law on inertia in our daily life

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths