Evaluate \([i^{18}+(\frac{1}{i})^{25}]^3.\)
Solution and Explanation
\([i^{18}+(\frac{1}{i})^{25}]^3\)
=\([i^{4×4+2}+(\frac{1}{i^{4×4+1}})]^3\)
=\([i^{(4)^4.i^2}+\frac{1}{(i^{4})^6.i}]^3\)
\(=[i^2+\frac{1}{i}]^3\) \([i^{-4}=1]\)
\(=[-1+\frac{1}{i}×\frac{1}{i}]^3\) \([i^4=-1]\)
\(=[-1+\frac{i}{i^2}]^3\)
\(=[-1-i]^3\)
=\((-1)^3[1+i]^3\)
\(=[1^3+i^3+3.1.i(1+i)]\)
\(=-[1+i^3+3i+3i^2]\)
\(=-[1+i^3+3i+3i^2]\)
\(=-[1-i+3i-3]\)
\(=-[-2+2i]\)
\(=2-2i\)
Top Questions on Algebra of Complex Numbers
Solve for \( x \):
\( \log_{10}(x^2) = 2 \).- BITSAT - 2025
- Mathematics
- Algebra of Complex Numbers
- Find the roots of the quadratic equation \( 2x^2 - 4x - 6 = 0 \).
- MHT CET - 2025
- Mathematics
- Algebra of Complex Numbers
- If the roots of the quadratic equation \( x^2 - 5x + 6 = 0 \) are \( p \) and \( q \), then what is the value of \( p + q \)?
- MHT CET - 2025
- Mathematics
- Algebra of Complex Numbers
Let \( K \) be an algebraically closed field containing a finite field \( F \). Let \( L \) be the subfield of \( K \) consisting of elements of \( K \) that are algebraic over \( F \).
Consider the following statements:
S1: \( L \) is algebraically closed.
S2: \( L \) is infinite.
Then, which one of the following is correct?- GATE MA - 2025
- Mathematics
- Algebra of Complex Numbers
- If the expression \( 7 + 6x - 3x^2 \) attains its extreme value \( \beta \) at \( x = \alpha \), then the sum of the squares of the roots of the equation \( x^2 + ax - \beta = 0 \) is:
- TS EAMCET - 2024
- Mathematics
- Algebra of Complex Numbers
Questions Asked in CBSE Class XI exam
Give reasons for the following.
(i) King Tut’s body has been subjected to repeated scrutiny.
(ii) Howard Carter’s investigation was resented.
(iii) Carter had to chisel away the solidified resins to raise the king’s remains.
(iv) Tut’s body was buried along with gilded treasures.
(v) The boy king changed his name from Tutankhaten to Tutankhamun.- CBSE Class XI
- Discovering Tut: The saga Continues
- \(\text{tan x}=-\frac{4}{3},\text{x\, in\, quadrant \,II.}\)
- CBSE Class XI
- Trigonometric Functions of Sum and Difference of Two Angles
Find the mean deviation about the median for the data
xi 15 21 27 30 35 fi 3 5 6 7 8 - CBSE Class XI
- Mean Deviation
- The Cartesian product A x A has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of A x A.
- CBSE Class XI
- Relations and functions
- ‘Have you come back?’ said the woman. ‘I thought that no one had come back.’ Does this statement give some clue about the story? If yes, what is it?
- CBSE Class XI
- The address
Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.