Advertisements
Advertisements
Question
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(1 + i)−3
Advertisements
Solution
(1 + i)−3 = `1/((1 + "i")^3`
= `1/(1 + 3"i" + 3"i"^2 + "i"^3)`
= `1/(1 + 3"i" - 3 - "i")` ...[∵ i2 = – 1, i3 = – i]
= `1/(-2 + 2"i")`
= `1/(-2 + 2"i") xx (-2 - 2"i")/(-2 - 2"i")`
= `(-2 - 2"i")/(4 - 4"i"^2)`
= `(-2 - 2"i")/(4 + 4)` ...[∵ i2 = – 1]
= `(-2 -2"i")/8`
= `-1/4 - 1/4 "i"`
This is of the form a + bi, where a = `-1/4` and b = `-1/4`
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
i457
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i49 + i68 + i89 + i110
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Find the value of the following expression:
(1 + i)6 + (1 − i)3
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write the argument of −i.
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
The polar form of (i25)3 is
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The principal value of the amplitude of (1 + i) is
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
The amplitude of \[\frac{1}{i}\] is equal to
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
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 + i)(1 − i)−1
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Evaluate the following : i93
