find the cube root of negative 729 3 square root of negative 729
The cube root of -729 can be found by evaluating (-729)^(1/3), which is equivalent to taking the cube root of 729 and then adding a negative sign since the number inside the root is negative.
The cube root of 729 is 9, so (-729)^(1/3) equals -9.
To find the cube root of a number, including negative numbers, you can follow these steps:
1. Rewrite the number in the form of a complex number: -729 = -729 + 0i
The complex number form consists of two parts: the real part and the imaginary part. In this case, the imaginary part is zero as the number is negative.
2. Convert the complex number to its trigonometric form:
-729 + 0i can be expressed as r(cosθ + isinθ), where r is the modulus (absolute value) of the number and θ is the argument (angle) of the number.
To find the modulus, calculate the cube root of the absolute value of the number:
| -729 | = 729
cbrt(729) = 9
To find the argument, divide the angle by 3:
argument = 0° / 3 = 0°
Therefore, the trigonometric form of -729 is 9(cos0° + isin0°).
3. Use De Moivre's theorem to find the cube root of the complex number:
De Moivre's theorem states that for any complex number z = r(cosθ + isinθ) and any positive integer n,
z^n = r^n (cos(nθ) + isin(nθ))
Applying De Moivre's theorem to the complex number -729, we have:
(-729)^(1/3) = 9^(1/3) [cos(0°/3) + isin(0°/3)]
= 9 [cos(0°) + isin(0°)]
= 9 [1 + i0]
= 9
Therefore, the cube root of -729 is 9.
For the second part of your question, finding the square root of a negative number does not yield a real number. The square root of -729 cannot be expressed as a real number but can be expressed as a complex number. However, if you are looking for the principal square root, it can be denoted as √(-729) = √729i, where i is the imaginary unit (√(-1)).