Advertisements
Advertisements
Question
Evaluate the following:
\[\frac{1}{i^{58}}\]
Advertisements
Solution
`1/i^58 = 1 /(i^(4 xx 14 +2)`
\[ = \frac{1}{\left( i^4 \right)^{14} \times i^2}\]
\[ = \frac{1}{i^2} \left( \because i^4 = 1 \right)\]
\[ = - 1 \left( \because i^2 = - 1 \right)\]
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 – i) – (–1 + i6)
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)`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Find the value of the following expression:
i49 + i68 + i89 + i110
Find the value of the following expression:
i30 + i80 + i120
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:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
If z1 is a complex number other than −1 such that \[\left| z_1 \right| = 1\] and \[z_2 = \frac{z_1 - 1}{z_1 + 1}\] then show that the real parts of z2 is zero.
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
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 sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
The principal value of the amplitude of (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
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
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:
`((1 + "i")/(1 - "i"))^2`
Evaluate the following : i116
Evaluate the following : `1/"i"^58`
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
State True or False for the following:
The order relation is defined on the set of complex numbers.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
Show that `(-1+ sqrt(3)i)^3` is a real number.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
Show that `(-1+sqrt3i)^3` is a real number.
