Question

Find the cube roots of : \[ - 1\].
\[\left( 1 \right)\] \[1\]
\[\left( 2 \right)\] \[ - 1\]
\[\left( 3 \right)\] \[\sqrt { - 1} \]
\[\left( 4 \right)\] Cannot be determined

Answer 1

Hint: We have to find the value of the cube root of \[ - 1\]. We solve this question using the concept of solving the linear equations. We should also have the knowledge of the formula of the sum of cubes of two numbers. First we will make an expression in terms of cubes of the variable and \[ - 1\], then using the formula of sum of cubes of two numbers we will split the expression and then we will solve the expression using the method of solving the quadratic equations. Using the formula, we can find the cube roots of the given number.

Complete step-by-step solution:
Given :
Cube root of \[ - 1\]
Let us consider that \[x\] is the cube root of the given number, so we can write the expression as :
\[x = {\left( { - 1} \right)^{\dfrac{1}{3}}}\]
Taking cube, we get the expression as :
\[{x^3} = - 1\]
\[{x^3} + 1 = 0\]
Now, we know that the sum of cubes of two numbers is given as :
\[{a^3} + {b^3} = (a + b) \times ({a^2} + {b^2} - ab)\]
Now, using the formula of the sum of cubes of numbers, we can write the expression as :
\[(x + 1) \times ({x^2} + {1^2} - x \times 1) = 0\]
\[(x + 1) \times ({x^2} + 1 - x) = 0\]
And using the expression, we get
\[\left( {x + 1} \right) = 0\] or \[({x^2} + 1 - x) = 0\]
Solving the first expressions, we get the value of \[x\] as :
\[x = - 1\]
Now, we will find the value of \[x\] for the second expression using the formula of quadratic equations.
We know that the formula for the roots of the quadratic equation \[a{x^2} + bx + c = 0\] is given as :
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
Using the formula, we get the value of \[x\] as :
\[x = \dfrac{{ - \left( { - 1} \right) \pm \sqrt {{{\left( { - 1} \right)}^2} - 4 \times 1 \times 1} }}{{2 \times 1}}\]
\[x = \dfrac{{1 \pm \sqrt {1 - 4} }}{2}\]
We get the value of \[x\] as :
\[x = \dfrac{{1 \pm 3i}}{2}\]
Where \[i = \sqrt { - 1} \].
Hence, we get the value of \[x\] as \[ - 1\], \[\dfrac{{1 + 3i}}{2}\] and \[\dfrac{{1 - 3i}}{2}\].
Thus, the correct option is \[\left( 2 \right)\].

Note: We can also solve this question using the concept of the theorem of De Moivre’s Theorem. We also use the concept of argument and modulus functions. We should also have the knowledge of writing a complex number into its polar form. First we will consider the given expression of a complex number to a variable. Then using the argument of a complex number we will write the polar form of the given complex expression and then we will take the power as the cube root and then using De Moivre’s Theorem we will find the value of the cube root of \[ - 1\].