Advertisements
Advertisements
Question
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Advertisements
Solution
`(1/3 + 3i)^3 = (1/3)^3 + 3 (1/3)^2 (3i) + 3(1/3) (3i)^2 + (3i)^3`
= `1/27 + 27i^3 + 9i^2`
= `1/27 + 27 (-i) + i + 9 (-1)`
= `1/27 - 9 + i (1 - 27)`
= `(1 - 243)/27 + i (-26)`
= `(- 242)/27 - 26i`
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
(ii) i528
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
If z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
Write −1 + i \[\sqrt{3}\] in polar form .
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
Which of the following is correct for any two complex numbers z1 and z2?
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + i)(1 − i)−1
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i888
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
