Calculating \((1+i)^4 + (1-i)^4\)
This problem involves calculating the sum of powers of complex numbers. We need to find the value of the expression \((1+i)^4 + (1-i)^4\), where the imaginary unit \(i\) is defined as \(\sqrt{-1}\).
Evaluating \((1+i)^4\)
We can evaluate \((1+i)^4\) in a couple of ways. One common method is to first calculate \((1+i)^2\).
- \((1+i)^2 = 1^2 + 2(1)(i) + i^2\)
- Since \(i^2 = -1\), this becomes \(1 + 2i - 1\).
- Simplifying, we get \((1+i)^2 = 2i\).
Now, we can find \((1+i)^4\) by squaring the result of \((1+i)^2\):
- \((1+i)^4 = ((1+i)^2)^2\)
- Substituting the value we found: \((1+i)^4 = (2i)^2\).
- Squaring this gives \(4i^2\).
- Again, since \(i^2 = -1\), we have \(4(-1)\).
- Therefore, \((1+i)^4 = -4\).
Evaluating \((1-i)^4\)
Similarly, we evaluate \((1-i)^4\). First, let's find \((1-i)^2\).
- \((1-i)^2 = 1^2 - 2(1)(i) + i^2\)
- Substituting \(i^2 = -1\): \(1 - 2i - 1\).
- Simplifying, we get \((1-i)^2 = -2i\).
Now, we find \((1-i)^4\) by squaring \((1-i)^2\):
- \((1-i)^4 = ((1-i)^2)^2\)
- Substituting the value we found: \((1-i)^4 = (-2i)^2\).
- Squaring this gives \(4i^2\).
- Since \(i^2 = -1\), this is \(4(-1)\).
- Therefore, \((1-i)^4 = -4\).
Summing the Results
Finally, we need to add the results of \((1+i)^4\) and \((1-i)^4\) together.
- The expression is \((1+i)^4 + (1-i)^4\).
- Substituting the values we calculated: \((-4) + (-4)\).
- The sum is \(-8\).
Conclusion
The value of \((1+i)^4 + (1-i)^4\) is \(-8\). This corresponds to the option provided.
Similar Questions
- If \(z\) is any complex number and \(iz^3 + z^2 - z + i = 0\), where \(i = \sqrt{-1}\), then what is the value of \((|z|+1)^2\)?
- Let \(z_1\) and \(z_2\) be two complex numbers such that \(\left|\frac{z_1+z_2}{z_1-z_2}\right| = 1\), then what is \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\) equal to ?
- If \(\omega \ne 1\) is a cube root of unity, then what are the solutions of \((z-100)^3 + 1000 = 0\) ?
- If \(\omega \neq 1\) is a cube root of unity, then what is \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) equal to?
- What is the value of the sum
\(\sum_{n=1}^{20}(i^{n-1} + i^n + i^{n+1})\)
where \(i = \sqrt{-1}\)? - If \(x, y\) and \(z\) are the cube roots of unity, then what is the value of \(xy + yz + zx\)?
- If \(\alpha\), \(\beta\), \(\gamma\) are cube roots of -8, then what is \(\frac{\alpha^2 p^2 + \beta^2 q^2 + \gamma^2 r^2}{\beta^2 p^2 + \gamma^2 q^2 + \alpha^2 r^2}\) equal to ?
- What is the value of \(\left|\frac{Z_1}{Z_2}\right|\)?
- What is the value of \(\frac{1}{2} + \text{Re}\left(\frac{Z_1}{Z_2}\right)\)?
- Consider the following inequalities :
I. \(1 + 4i > 3 + 2i\)
II. \(2 + 3i < 3 + 4i\)
III. \(4 + 3i > 3 + 4i\)
where \(i = \sqrt{-1}\)
How many of the above are valid ?
Important Questions from Complex Numbers
If \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\), then the value of x2 + y2 is -
If x = cos θ + i sin θ, then the value of \({x^n} + \frac{1}{{{x^n}}}\) is:
If \(\left| {\begin{array}{*{20}{c}} {6i}&{ - 3i}&1\\ 4&{3i}&{ - 1}\\ {20}&3&i \end{array}} \right| = x + iy\), then the values of x and y are:
If iz3 + z2 - z + i = 0, then the value of |z| is:
Nature of the triangle formed by the points representing the complex numbers 3 + 4i, 8 - 6i and 13 + 9i is:
