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MissP.25_5 said:Hello everyone.
How to find the 4th root of -4? I know it's just plugging in the number into the formula but how since n=4, how can we calculate that without calculator? And how to draw it? Here I attached what I have done so far.
SteamKing said:You've made a simple mistake in your calculation of arg(-4).
z = -4 + i0, or (-4, 0)
θ = arctan (0/-4) = π
Even though z = -4, draw it on the complex plane properly.
MissP.25_5 said:Oh yes, that was a careless mistake. Ok, so now I have the values:
r=4
n=4
θ=∏
But how do I find 4√4 ?
SteamKing said:Use this information and apply Euler's formula.
z = r e[itex]^{i θ}[/itex]
z[itex]^{1/n}[/itex] = r[itex]^{1/n}[/itex] e[itex]^{i kθ / n}[/itex], [itex]0\leq k \lt n[/itex]
See:
http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Roots.aspx
HallsofIvy said:You know that [tex]x^4= (x^2)^2[/tex], right? So [tex]\sqrt[4]{4}= \sqrt{\sqrt{4}}[/tex]. What is the square root of 4? What is the square root of that?
MissP.25_5 said:If I compute z = r e[itex]^{i θ}[/itex], wouldn't that bring us back to the start? Because that is -4.
Could you elaborate please? I don't really get it.
SteamKing said:Use this information and apply Euler's formula.
z = r e[itex]^{i θ}[/itex]
z[itex]^{1/n}[/itex] = r[itex]^{1/n}[/itex] e[itex]^{i kθ / n}[/itex], [itex]0\leq k \lt n[/itex]
The 4th root of -4 is -1.41421. This is because when a number is raised to the 4th power, the result is the number multiplied by itself four times. So, the 4th root of -4 is the number that, when multiplied by itself four times, equals -4.
To calculate the 4th root of -4, you can use a calculator or manually use the exponent and root functions. To manually calculate, you can raise -4 to the power of 1/4, which is the same as taking the 4th root of -4. The result is -1.41421.
Yes, the 4th root of -4 can be simplified to -2. This is because the 4th root of a negative number can be written as the product of the square root of the number and the 4th root of -1. So, the 4th root of -4 is equal to the square root of 4 times the 4th root of -1, which simplifies to -2.
The 4th root of -4 is represented on a graph as a point on the x-axis at -1.41421. This is because the x-coordinate of a point on the x-axis is the same as the value of the number. So, the 4th root of -4 is represented by a point at -1.41421 on the x-axis.
The 4th root of -4 has a real part of -1.41421 and an imaginary part of 0. This is because the 4th root of -4 is a complex number, which has both a real and imaginary part. The real part represents the number on the x-axis and the imaginary part represents the number on the y-axis. Since the 4th root of -4 lies on the x-axis, its imaginary part is 0.