Advertisements
Advertisements
Question
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
Advertisements
Solution
`((-1 + sqrt(-3))/2)^3 = ((-1 + sqrt(3 xx - 1))/2)^3`
= `((-1 + sqrt(3)"i")/2)^3`
= `(-1 + sqrt(3)"i")^3/8`
= `1/8[(-1)^3 + 3(-1)^2 sqrt(3)"i" + 3(-1)(sqrt(3)"i")^2 + (sqrt(3)"i")^3]`
= `1/8[-1 + 3sqrt(3)"i" - 3 xx 3"i"^2 + 3sqrt(3)"i"^3]`
= `1/8[-1 + 3sqrt(3)"i" + 9 - 3sqrt(3)"i"]` ...[∵ i2 = – 1, i3 = – i]
= `1/8(8)`
= 1, which is a rational number.
APPEARS IN
RELATED QUESTIONS
Find the multiplicative inverse of the complex number.
`sqrt5 + 3i`
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
`5/2"i"(- 4 - 3 "i")`
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Simplify the following and express in the form a + ib:
`(1 + 2/i)(3 + 4/i)(5 + i)^-1`
Find the value of i49 + i68 + i89 + i110
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
Select the correct answer from the given alternatives:
`sqrt(-3) sqrt(-6)` is equal to
Answer the following:
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
The value of (2 + i)3 × (2 – i)3 is ______.
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
State true or false for the following:
The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.
State true or false for the following:
If n is a positive integer, then the value of in + (i)n+1 + (i)n+2 + (i)n+3 is 0.
What is the value of `(i^(4n + 1) -i^(4n - 1))/2`?
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
What is the reciprocal of `3 + sqrt(7)i`.
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
If `((1 + i)/(1 - i))^x` = 1, then ______.
If a + ib = c + id, then ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.
The complex number z = x + iy which satisfy the equation `|(z - 5i)/(z + 5i)|` = 1, lie on ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals 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 `(i^592+i^590+i^588+i^586+i^584)/(i^582+i^580+i^578+i^576+i^574)`
