Solving x^4-16=0 

Solve x^3=8 ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-7ac4fp7M4t0.html

ί told π, "be rational!" π told ί, "get real!"

How about using the rule of *Difference of (4th) power* ? x⁴-y⁴ = (x-y)(x³+x²y+xy²+y³) If the equation must be solved in this way , then what should we do further?

there are n valid roots to a radical of degree n. the main one we think of, the principal root, tends to be the positive root for real number inputs, but for complex inputs the principal root is the one with 1/nth the angle with the positive real axis compared to the input. if you look at some stuff about "roots of unity" youll see these roots are equally spaced in the complex plane, and are all of the same magnitude. we want the fourth roots of 16, so there are four roots that divide the complex plane into four. the principal root is positive because the input is real, which means the others must also land on axes, and one could tell reasonably that 16 is the fourth power of 2, as its an important sequence. so altogether we can deduce that the solutions are 2, 2i, -2, and -2i in that order. all can be multiplied by themselves four times to reach 16, as expected.

I did not know that a^2 + b^2 = (a + bi)(a - bi), thanks!!

Curious why you don't factor that with your "tic tac toe" method? You will quite easily get the conjugates and the formula just factoring without memorizing the difference of squares.

2

I thought x was 2 until I realised this ain't algebra

Can't you just simplify it to x^2=4 then x=+-2?

You are making this way more complicated than it needs to be

I think you posted this to the wrong channel since you're taking us to the imaginary world.

If we stand defiantly claiming only rational solutions matter what do we get? LOL.

I still don't understand why it's equal to zero... Doesn't that make the whole equation pointless?

5:33 how did you do that!

Am I the only person who watched math videos of things I already know how to do, because he finds it kinda comforting?

You had me until i

I don't remember if I learned this method but my first thought was square rooting both sides by four which I would only get two from.

Would this be an acceptable answer? x=±√(±√(16))

You can also do it with substitution: x⁴ = 16 Let z ≔ x²: z² = 16 z = ±4 Resubstitution: 4 = x² ⇒ x = ±2 and -4 = x² ⇒ x = ±2i

It might be interesting to talk about how in a simple situation like this, all the answers are equally spaced points on a circle. This had helped my get a handle on what the answers ought to look like, and sanity check for the range of values expected.

If you came from my IG 👇

The WHOLE solution is easier. Just Take the 4th root from each side and you get 2😂

You inspire me and make me love math even more.

x^4=16 x=16^1/4 x=4√16 x=±2

i know nothing of maths compared to this

x1=±2 x2=±2i

This is one of those math questions where I knew the answer. But don't ask me how I knew. I can't explain 'i'. All the parentheses confused me. I just know the answer

Yeah once you started adding i’s, this is where i get it wrong. I never learned adding i’s.

Wow! I never thought of it that way.❤😊

imagine that

Imaginary numbers are weird, but they calculate the negative space within a square or cube. It’s essentially what is missing.

Me : I only want answers that matter, so I'm tossing out any imaginary solutions, and even negative solutions because I'm using this equation to represent the real world and everything I care about is to the right of the origin. Punches number into calculator uses the "x-root" button (or raise to power of 1/4) calls it a day.

Step 1: Be familiar enough with binary to already know the answer at a glance.

I like using the geometry to explain it. I did not originally learn that trick.

You can also use demoivres to make sure you never forget abt the complex numbers

Why can I watch this video in PiP mode? I didn't even know this existed

He’s brought i to math basics ono

Do you guys know who is dat in photo frame?

Smash x next question

first comment!love ur videos.

Cool!

Why aren't there an infinite number of solutions?

What happened at the end? Why the sad, dismayed, face?!? Is everything okay?

From (X²)² - (4)² = 0 could you just add (4)² to each side and take the square root and get to the same answer faster without factoring?

Woah, I know this topic but somehow u made it way more confusing than it needs to be? e.g. (a - b) (a + b) =a^2 - b^2 because: (a - b) (a + b) = a(a - b) + b(a - b) (a - b) (a + b) = a^2 - ab + ab -b^2 (a - b) (a + b) = a^2 - b^2 Simple as that, the square explanation was way too confusing for me atleast

Solution: x⁴ = 16 (x²)² = 16 |√ |x²| = 4 Two cases: x² ≥ 0 and x² < 0 Case x² ≥ 0: x² = 4 |√ |x| = 2 Two cases: x ≥ 0 and x < 0 x₁ = 2 -x₂ = 2 → x₂ = -2 Case x² < 0: -x² = 4 |*-1 x² = -4 |√ |x| = 2i Two cases: x ≥ 0 and x < 0 x₃ = 2i -x₄ = 2i → x₄ = -2i Therefore x ∈ {-2, 2, -2i, 2i}

√2 , next question