Advertisements
Advertisements
Question
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Solution
`(1 + "i")^4 xx (1 + 1/"i")^4`
= `[(1 + "i")(1 + 1/"i")]^4`
= `[(1 + "i") ((1 + "i"))/"i"]^4`
= `[((1 + "i")^2)/"i"]^4`
= `(1 + 2"i" + "i"^2)^4/"i"^4`
= `(1 + 2"i" - 1)^4/1` ...[∵ i2 = – 1]
= 16i4
= 16 ...[∵ i4 = 1]
APPEARS IN
RELATED QUESTIONS
Express the following expression in the form of a + ib.
`((3 + sqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Find the value of i49 + i68 + i89 + i110
Simplify the following and express in the form a + ib:
(2 + 3i)(1 – 4i)
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Write the conjugates of the following complex number:
`-sqrt(-5)`
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Find the value of i49 + i68 + i89 + i110
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
Answer the following:
Simplify the following and express in the form a + ib:
(1 + 3i)2(3 + i)
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
The value of (2 + i)3 × (2 – i)3 is ______.
What is the reciprocal of `3 + sqrt(7)i`.
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
If the real part of `(barz + 2)/(barz - 1)` is 4, then show that the locus of the point representing z in the complex plane is a circle.
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).
The value of `(z + 3)(barz + 3)` is equivalent to ______.
If z is a complex number, then ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.
If α, β, γ and a, b, c are complex numbers such that `α/a + β/b + γ/c` = 1 + i and `a/α + b/β + c/γ` = 0, then the value of `α^2/a^2 + β^2/b^2 + γ^2/c^2` is equal to ______.
If `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18`
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Simplify the following and express in the form a+ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Evaluate the following:
i35
