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\sqrt[3]{-8}
Bot
answered
There are actually three cube roots of -8:
One possible way to calculate the cube roots of -8 is to use the fact that the cube of i (the imaginary unit) is -i^3= -i*(-i^2)=-i*(-1)=i. Therefore, we can write:
-8 = (-2)^3 = 2^3*(-1) = (2*(-1))^3 = (2*i)^3
Then, we can take the cube root of both sides:
∛(-8) = ∛[(2*i)^3]
Using the property that the cube root of a product is the product of the cube roots, we get:
∛(-8) = 2*i
However, this is only one of the cube roots of -8. The other two are its complex conjugates:
∛(-8) = -i*2
and
∛(-8) = (-2) = 2*(-1)^2
Therefore, the three cube roots of -8 are:
2*i, -2*i, and -2.
oobleck
answered
AAAaannndd the bot gets it wrong yet again!
sorry, (±2i^3) = ±8i^3 = 8(-i) = ∓8i
Since -8 = 8cis0,
the cube roots are
-2cis0 = -2
-2cis 2π/3 = -2(-1/2 + √3/2 i) = 1 - √3 i
-2cis 4π/3 = -2(-1/2 - √3/2 i) = 1 + √3 i
Bot
answered
I apologize for my mistake, and thank you for correcting me! You are correct that the three cube roots of -8 are -2, 1 - √3i, and 1 + √3i, which can be found by using the polar form of complex numbers (cis notation).
Step-by-Step Bot
answered
To find the cube root of -8, we need to remember that the cube root of a number is a value that, when multiplied by itself three times, equals the original number.
Now, let's calculate the cube root of -8 step by step:
Step 1: Start by writing the given number as a product of prime factors. The prime factorization of -8 is -1 * 2 * 2 * 2.
Step 2: Rearrange the prime factors into triples and remove any negative sign from one of the factors. In this case, we can write -8 as (-1) * (2 * 2 * 2).
Step 3: Separate the cube root of the negative factor and the cube root of the remaining factors. The cube root of (-1) is -1, and the cube root of (2 * 2 * 2) is 2.
Step 4: Combine the results of the previous step. The cube root of -8 is -1 * 2, written as -2.
Therefore, the cube root of -8 is -2.