I just need help with these questions, thanks.
i^5
3(-2i)^2
-i(isqrt5)^2
Remember that i represents the square root of (-1). So, i^2 = -1
i^5 = (i^2)(i^2)(i)
=(-1)(-1)(i)
=?
Remember that i^2 = -1
3(-2i)^2
=3(-2)^2(i^2)
=?
-i(isqrt(5))^2
=-i(i^2)(sqrt(5))^2
=?
Sure! Let's break down each of the complex number expressions and simplify them step by step.
1. i^5
To solve this, we need to understand the pattern of powers of i. The powers of i cycle every four terms: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. From here, the pattern repeats itself.
To calculate i^5, we can write it as i^4 * i^1. This gives us (1) * i = i. Therefore, i^5 = i.
2. 3(-2i)^2
First, we need to square the expression inside the parentheses. (-2i)^2 can be calculated by multiplying the number by itself:
(-2i)^2 = (-2i) * (-2i) = 4i^2.
Now, we can substitute i^2 with -1, according to the pattern mentioned earlier:
4i^2 = 4(-1) = -4.
Finally, multiply the result by 3:
3(-2i)^2 = 3 * (-4) = -12.
Therefore, 3(-2i)^2 simplifies to -12.
3. -i(isqrt5)^2
Similar to the previous example, we first square the expression in the parentheses:
(isqrt5)^2 = (isqrt5) * (isqrt5) = -5.
Now, we multiply -i by -5:
-i(isqrt5)^2 = -i * (-5) = 5i.
Thus, -i(isqrt5)^2 simplifies to 5i.