Advertisements
Advertisements
Question
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
Advertisements
Solution
`((1 + "i")/sqrt(2))^2 = (1 + 2"i" + "i"^2)/2 = (1 + 2"i" - 1)/2` = i
∴ `((1 + "i")/sqrt(2))^8 = [((1 + "i")/sqrt(2))^2]^4` = i4 = 1 .......(i)
Also, `((1 - "i")/sqrt(2))^2 = (1 - 2"i" + "i"^2)/2 = (1 - 2"i" - 1)/2` = – i
∴ `((1 - "i")/sqrt(2))^8 = [((1 - "i")/sqrt(2))^2]^4`
= (– i)4 = (– 1)4 × (i)4
= 1 × i4
= 1 ........(ii)
Adding (i) and (ii), we get
`((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 1 + 1 = 2
APPEARS IN
RELATED QUESTIONS
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
`-sqrt(-5)`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Show that 1 + i10 + i100 − i1000 = 0
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
Select the correct answer from the given alternatives:
If n is an odd positive integer then the value of 1 + (i)2n + (i)4n + (i)6n is :
Select the correct answer from the given alternatives:
The value of is `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)` is equal to:
Answer the following:
Simplify the following and express in the form a + ib:
(2i3)2
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
Evaluate: (1 + i)6 + (1 – i)3
Find the value of k if for the complex numbers z1 and z2, `|1 - barz_1z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |"z"_2|^2)`
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
State true or false for the following:
If n is a positive integer, then the value of in + (i)n+1 + (i)n+2 + (i)n+3 is 0.
What is the principal value of amplitude of 1 – i?
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
If |z + 1| = z + 2(1 + i), then find z.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
State True or False for the following:
The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).
Find `|(1 + i) ((2 + i))/((3 + i))|`.
If z1, z2, z3 are complex numbers such that |z1| = |z2| = |z3| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z1 + z2 + z3| is ______.
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.
If `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
Find the value of `(i^592 + i^590 + i^588 + i^586 + i^584)/ (i^582 + i^580 + i^578 + i^576 + i^574)`
