Advertisements
Advertisements
Question
Show that `(-1 + sqrt(3)"i")^3` is a real number
Advertisements
Solution
`(-1 + sqrt(3)"i")^3`
`= (-1)^3 + 3(-1)^2 (sqrt(3)"i") + 3(-1) (sqrt(3)"i")^2 + (sqrt(3"i"))^3` ...[(a + b)3 = a3 + 3a2b + 3ab2 + b3]
= `-1 + 3sqrt(3)"i" - 3(3"i"^2) + 3sqrt(3)"i"^3`
= `-1 + 3sqrt(3)"i" - 3(-3) - 3sqrt(3)"i"` ...[∵ i2 = – 1, i3 = – i]
= `-1 + 3sqrt(3)"i" + 9 - 3sqrt(3)"i"` ...[∵ i2 = – 1, i3 = – i]
= – 1 + 9
= 8, which is a real number
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Express the given complex number in the form a + ib: (1 – i)4
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
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:
i + i2 + i3 + i4
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
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`
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The principal value of the amplitude of (1 + i) is
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
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
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Find a and b if abi = 3a − b + 12i
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`("i"(4 + 3"i"))/((1 - "i"))`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
Evaluate the following : i93
Evaluate the following : `1/"i"^58`
Show that 1 + i10 + i20 + i30 is a real number
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.
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
State True or False for the following:
2 is not a complex number.
Show that `(-1+ sqrt(3)i)^3` is a real number.
