Please help me with this problem.
Simplify.
√(-24 - 10i)
oh, and also, if it is not too much too ask, please include steps and explanations. ty
√(-24 - 10i)
If you want to find a perfect square which equals -24-10i, it will have to include -5i, so
(a-5i)^2 = -24-10i
a^2 - 10ai - 25 = -24-10i
Hmmm. Looks like a=1
(1-5i)^2 = 1 - 10i - 25 = -24-10i
So,
√(-24 - 10i) = 1-5i or -1+5i
Or, if you use polar form, you can see that
r=26
tanθ = 5/12
the square root has
r = √26
tanθ = (1-(-12/13))/(-5/13) = -5
so, we get the same result.
Class 11
To simplify the expression √(-24 - 10i), we first need to determine the square root of the complex number.
Let's break down the process into multiple steps:
Step 1: Rewrite the complex number in the form a + bi.
In this case, -24 - 10i is already in the form a + bi.
Step 2: Calculate the magnitude (modulus) of the complex number.
The magnitude of a complex number z = a + bi can be found using the formula √(a^2 + b^2).
In our case, the magnitude of √(-24 - 10i) is √((-24)^2 + (-10)^2).
Simplifying further, we get √(576 + 100) = √676 = 26.
Step 3: Find the argument (angle) of the complex number.
The argument of a complex number z = a + bi can be found using the formula tan^-1(b/a).
In our case, the argument of √(-24 - 10i) is tan^-1(-10/-24).
Simplifying further, we get tan^-1(10/24) = tan^-1(5/12).
Note: Since we are dealing with a square root (√), the argument will be divided by 2.
Step 4: Express the simplified form of the root.
Now, we can express the simplified form using the magnitude and argument we calculated:
√(-24 - 10i) = √26 * cis(tan^-1(5/12)).
In this form, cis(θ) represents the complex number with an argument of θ in trigonometric form.
So, the simplified answer is √26 * cis(tan^-1(5/12)).