Answer
Verified
409.5k+ views
Hint:To find the value of the term we use the complex number multiplication expressed in form of the complex number as \[a+ib\] where the variable a and b are real numbers with the value of \[i\] as imaginary unit, when multiplying the value of \[i\times i\] we get the result of product as \[-1\] which can be used in the above question.
Complete step by step solution:
Now as given in the question, the term \[\dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}\] first need to simplify it in terms of \[i\] and constant numbers. Expanding the numerator and denominator in terms of simpler complex numbers, we get the numerator as:
\[\Rightarrow {{\left( 1-i \right)}^{3}}=1-i-3i+3{{i}^{2}}\]
And the denominator is written as:
\[\Rightarrow 1-{{i}^{3}}=1+i\]
Now placing the expanding part of the numerator and the denominator we get the term as:
\[\Rightarrow \dfrac{1-\left( -i \right)-3i+3{{i}^{2}}}{1+i}\]
\[\Rightarrow \dfrac{-2-2i}{1+i}\]
\[\Rightarrow \dfrac{-2\left( 1+i \right)}{1+i}\]
\[\Rightarrow -2\]
Therefore, the value of the term \[\dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}\] is \[-2\].
Note: Another method to solve the question is by:
\[\Rightarrow \dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}=\dfrac{\left( 1-i \right)\left( 1-i \right)\left( 1-i \right)}{1-1\times -i}\]
\[\Rightarrow \dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}=\dfrac{\left( 1-i \right)\left( 1-i \right)\left( 1-i \right)}{1+i}\]
\[\Rightarrow \dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}=\dfrac{-2\left( 1+i \right)}{1+i}\]
\[\Rightarrow \dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}=-2\]
Complete step by step solution:
Now as given in the question, the term \[\dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}\] first need to simplify it in terms of \[i\] and constant numbers. Expanding the numerator and denominator in terms of simpler complex numbers, we get the numerator as:
\[\Rightarrow {{\left( 1-i \right)}^{3}}=1-i-3i+3{{i}^{2}}\]
And the denominator is written as:
\[\Rightarrow 1-{{i}^{3}}=1+i\]
Now placing the expanding part of the numerator and the denominator we get the term as:
\[\Rightarrow \dfrac{1-\left( -i \right)-3i+3{{i}^{2}}}{1+i}\]
\[\Rightarrow \dfrac{-2-2i}{1+i}\]
\[\Rightarrow \dfrac{-2\left( 1+i \right)}{1+i}\]
\[\Rightarrow -2\]
Therefore, the value of the term \[\dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}\] is \[-2\].
Note: Another method to solve the question is by:
\[\Rightarrow \dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}=\dfrac{\left( 1-i \right)\left( 1-i \right)\left( 1-i \right)}{1-1\times -i}\]
\[\Rightarrow \dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}=\dfrac{\left( 1-i \right)\left( 1-i \right)\left( 1-i \right)}{1+i}\]
\[\Rightarrow \dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}=\dfrac{-2\left( 1+i \right)}{1+i}\]
\[\Rightarrow \dfrac{{{\left( 1-i \right)}^{3}}}{1-{{i}^{3}}}=-2\]
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
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
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE