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The 3 cube roots of a cubic equation are often going to require complex numbers, and this is one of them. The primary root is (-2) since (-2)3= -8 The other two roots are going to have the same mod (2) but be evenly spaced around the complex plane, so (2pi)/3 angular difference.
These angles are pi/3 and 5pi/3
(2,pi/3) = (1 + √3i) = (1 + 1.732i)
(2,5pi/3) = (1 + -√3i) = (1- 1.732i)
Check
(-2)3 = -8◄
(1 + √3i)3 = -8◄
(1 - √3i)3 = -8◄
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What are three cube roots of 8?
The cubic root of 8 = 2 2 * 2 * 2 = 8
What is the definition of a cube root?
The cube root for a real number is the value that when multiplied by itself three times produces that number. So that the cube root of B is A where A x A x A = B.Examplesthe cube root of 8 is 2... (2x2x2=8)the cube root of 27 is 3...(3x3x3=27)The special cases are 0 and 1, which are by definition their own roots (square roots, cube roots, etc.) because all powers of 0 and 1 are still the same value, zero or 1.Perfect cubes are numbers that can be cube-rooted into an integer.These include the numbers 0, 1, 8, 27, and so forth.Negative numbers can also be perfect cubes.-1x-1x-1=-1-2x-2x-2=-8-3x-3x-3=-27For non-perfect cubes, the root includes a repeating decimal requiring rounding.
Why does a cube root only have one value?
Because the cube of a positive number is positive and the cube of a negative number is negative.-------------------------------------------------------------------------------------------------------------------------------Every number has THREE cube roots. However, (at least) two of the three are complex numbers.For example, the cube roots of 8 are 2, (-1 + √3 i) and (-1 - √3 i) with i² = -1:2³ = 2 × 2 × 2 = 8(-1 + √3 i)³ = (-1 + √3 i)(-1 + √3 i)(-1 + √3 i)= (-1 + √3 i)((-1)² - 2√3 i + 3i²)= (-1 + √3 i)(1 - 2√3 i -3)= (-1 + √3 i)(-2 - 2√3 i)= (-1 + √3 i)(-1 - √3 i)2= ((-1)² - 3i²)2= (1 + 3)2= 4 × 2 = 8(-1 - √3 i)³ = (-1 - √3 i)(-1 - √3 i)(-1 - √3 i)= (-1 - √3 i)((-1)² + 2√3 i + 3i²)= (-1 - √3 i)(1 + 2√3 i -3)= (-1 - √3 i)(-2 + 2√3 i)= (-1 - √3 i)(-1 + √3 i)2= ((-1)² - 3i²)2= (1 + 3)2= 4 × 2 = 8
Which part of a cube is a vertex?
A vertex of a cube is its corner points. A cube has 8 vertices and six faces. It also has 12 edges. A cube is a three-dimensional figure.
Why do square roots have two solutions and cube roots only have one solution?
Square roots only have two solutions if the number is positive, if the number is negative it has no solutions.Actually ALL numbers (including negative numbers) have two square roots and three cube roots - just not all of them are real numbers; some are complex numbers.What is a complex number?A complex number is a number which is the sum of two parts: a real part and an imaginary part, ie are of the form (a + bi):a is the real part;bi is the imaginary part which is a real number (b) multiplied by the square root of -1 which is an imaginary value. To avoid having to write √-1 all the time, the little 'i' is used instead, ie i² = -1.Now the square roots of negative numbers can be found:eg √-4 = √(4 × -1) = √4 √-1 = 2i → the square root of -4 is ±2i.What about cube roots?³√8 = 2 is the real root, but there are two further complex roots: (-1 + √3 i) and (-1 - √3 i). I'll cube the first to show it does indeed equal 8 (remember that i² = -1)(-1 + √3 i)³ = (-1 + √3 i)(-1 + √3 i)(-1 + √3 i)= (-1 + √3 i)((-1)×(-1) + -1 × √3 i + -1 × √3 i + (√3 i)×(√3 i))= (-1 + √3 i)(1 - 2 (√3 i) + 3i²)= (-1 + √3 i)(1 - 2 (√3 i) + 3 × -1)= (-1 + √3 i)(1 - 2 (√3 i) - 3)= (-1 + √3 i)(-2 - 2 (√3 i))= (-1 + √3 i)(-1 - √3 i) × 2= ((-1)² - (√3 i)²) × 2= (1 - (-3)) × 2= (1 + 3) × 2= 4 × 2= 8Similar cubing of (-1 - √3 i) equals 8.
How do you find the cube root of a negative number?
If the negative number is "-a", then you can say the cube root is "-(cube root of a)" Because if you cube a negative number, you get a negative number. So if you cube root a negative number, you get a negative number. Ex) cube root of -8 = -2 Because (-2)^3 = -8 But if you want to find the complex cube roots, you can make an equation: "x^3=-a" or "x^3+a=0" We know one of the roots is "-(cube root of a)" so you can factor the equation by (x+(cube root of a)) And then you use the quadratic formula for the quadratic equation you're left with. Ex) x^3=-8 or x^3+8=0 Since -2 is a root, factor it by (x+2) x^3+8=(x+2)(x^2-2x+4) Using the quadratic formula, you get "1+i√3" and "1-i√3" Therefore the three cube roots of -8 is <"-2", "1+i√3", "1-i√3">
What are three cube roots of 8?
The cubic root of 8 = 2 2 * 2 * 2 = 8
What is the cube root of negative 8?
It is: -2
What are the positive and negative square roots of 64?
8 and -8.
How do you find the cube of a negative number?
The cube of a negative number is negative. Negative times negative is positive, but then you have the third multiplication, and positive times negative is negative. So for example (-2)3 = -8.
Why dont you have three answers for cube root?
You do. The other two are complex numbers, of interest only tomathematicians and engineers, and usually not listed.For example, the three cube roots of 8 are:2-1 + i sqrt(3)-1 - i sqrt(3)
How do you write cubed roots?
In mathematics, a cube root of a number, denoted or x1/3, is a number a such that a3 = x. All real numbers (except zero) have exactly one real cube root and a pair ofcomplex conjugate roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8 is 2, because 23 = 8. All the cube roots of −27iareThe cube root operation is not associative or distributive with addition or subtraction.The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers, for example:but
What is the cube root of -8 with complex answers?
-21 + 1.7320508i1 - 1.7320508i
How do you find the cube of 8?
Cube it. 8^3. Cubed means to the power of three.8x8x8=512
What is Cube Root of A Number?
it is the number which when multiplied by itself three times give you the original number. For example, the cube root of 8 is 2 (2x2x2 = 8) This works for negative numbers as well Forexample, the cube root of -27 is -3 [(-3)x(-3)x(-3) = -27]
What is the definition of a cube root?
The cube root for a real number is the value that when multiplied by itself three times produces that number. So that the cube root of B is A where A x A x A = B.Examplesthe cube root of 8 is 2... (2x2x2=8)the cube root of 27 is 3...(3x3x3=27)The special cases are 0 and 1, which are by definition their own roots (square roots, cube roots, etc.) because all powers of 0 and 1 are still the same value, zero or 1.Perfect cubes are numbers that can be cube-rooted into an integer.These include the numbers 0, 1, 8, 27, and so forth.Negative numbers can also be perfect cubes.-1x-1x-1=-1-2x-2x-2=-8-3x-3x-3=-27For non-perfect cubes, the root includes a repeating decimal requiring rounding.
Does negative eight plus negative three equal?
Negative eight (-8) plus negative three (-3) is equal to negative eleven (-11). It is the same as -8 - 3.
