Simplify: (1-i)^3
Here's how I break it down using the formula (x-y)^3 but I just don't know how to get to the final answer. Please help.
= 1-3i+3(i)^2-(i)^3
Thank you
so far, so good. Remember that
i^2 = -1
i^3 = -i
and you have
1 - 3i - 3 + i
-2 - 2i
your expansion is correct, except now you have to reduce the powers of i
look at this pattern
i^1 = i
i^2 = -1
i^3 = (i^2)i = -i
i^4 = (i^2)(i^2) = (-1)(-1) = 1
i^5 = (i^4)(i) = i
so it runs:
i , -1 , -i , +1 , i , -1 , -i , +1 , .....
notice that if the exponent is divisble by 4 i^n - = +1
if the exponent is even but not divisible by 4 i^n = -1
back to yours ...
1 - 3i + 3i^2 - i^3
= 1 - 3i - 3 + i
= -2 - 2i
Please explain why i^2=-1
by definition:
i = √-1
square both sides
i^2 = (√-1)(√-1) = -1
when you multiply √x by √x you get x
e.g. √48√48 = 48
√12.698√12.698 = 12.698
To simplify the expression (1-i)^3, you can expand it using the formula for the binomial expansion, (x-y)^3.
The formula for (x-y)^3 is:
(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
Now let's apply this formula to simplify (1-i)^3:
= 1^3 - 3 * 1^2 * i + 3 * 1 * i^2 - i^3
Simplifying further:
= 1 - 3i + 3i^2 - i^3
Now, we know that i is defined as the square root of -1. So, we can simplify i^2 as:
i^2 = -1
Replacing i^2 with -1 in the expression:
= 1 - 3i + 3 * (-1) - i^3
Now let's simplify i^3. Multiplying i^2 by i:
i^3 = i^2 * i = -1 * i = -i
Replacing i^3 with -i in the expression:
= 1 - 3i + 3 * (-1) - (-i)
Simplifying further:
= 1 - 3i - 3 + i
Combining like terms:
= -2 - 2i
So, the simplified expression is -2 - 2i.