Advertisements
Advertisements
Question
Express the following expression in the form of a + ib.
`((3 + sqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
Advertisements
Solution
`((3 + isqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
= `((3)^2 - (isqrt5)^2)/(sqrt3 + sqrt2i - sqrt3 + sqrt2i` `[(a+b) (a +b) = a^2 - b^2]`
= `(9 - 5i)^2/(2 sqrt2i)`
= `(9 - 5 (-1))/(2 sqrt2i)` `[i^2 = -1]`
= `(9 + 5)/(2 sqrt2i) xx i/i`
= `(14i)/((2sqrt2) (-1))`
= - `(7i)/sqrt2 xx sqrt2/sqrt2`
= `(-7 sqrt2i)/2`
APPEARS IN
RELATED QUESTIONS
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Show that 1 + i10 + i20 + i30 is a real number.
Simplify the following and express in the form a + ib:
(2i3)2
Find the value of: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.
Find the value of: x3 – 3x2 + 19x – 20, if x = 1 – 4i
Write the conjugates of the following complex number:
3 + i
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
Answer the following:
Evaluate: (1 − i + i2)−15
Answer the following:
Find the real numbers x and y such that `x/(1 + 2"i") + y/(3 + 2"i") = (5 + 6"i")/(-1 + 8"i")`
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
The value of (2 + i)3 × (2 – i)3 is ______.
Evaluate: (1 + i)6 + (1 – i)3
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
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.
What is the principal value of amplitude of 1 – i?
If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
Multiplicative inverse of 1 + i is ______.
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
If a + ib = c + id, then ______.
If the least and the largest real values of α, for which the equation z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution, are p and q respectively; then 4(p2 + q2) is equal to ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
Simplify the following and express in the form a + ib.
`(3i^5+2i^7+i^9)/(i^6+2i^8+3i^18)`
Find the value of `sqrt(-3) xx sqrt(-6)`
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.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
i2 + i3 + ... + i4000 =
