Answer
Verified
398.1k+ views
Hint:
Here, we will try to find the three cube roots of unity where one of them will be a real root and the other two will be imaginary roots. When we will add the squares of these three roots together, we will get a 0. Hence, by substituting the imaginary roots by omega and square of omega, we will be able to prove the given identity.
Formula Used:
$\left( {{a^3} - {b^3}} \right) = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)$
Determinant, $D = {b^2} - 4ac$
Quadratic formula, $z = \dfrac{{ - b \pm \sqrt D }}{{2a}}$
Complete step by step solution:
In order to prove the given identity,
Let us assume the cube root of unity or 1 as:
$\sqrt[3]{1} = z$
Cubing both sides, we get
$ \Rightarrow 1 = {z^3}$
Subtracting 1 from both the sides, we get
$ \Rightarrow {z^3} - 1 = 0$
Now, using the formula $\left( {{a^3} - {b^3}} \right) = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)$
$ \Rightarrow \left( {z - 1} \right)\left( {{z^2} + z + 1} \right) = 0$
Therefore, either $\left( {z - 1} \right) = 0$
$ \Rightarrow z = 1$
Or, $\left( {{z^2} + z + 1} \right) = 0$
Comparing with $\left( {a{x^2} + bx + c} \right) = 0$
Here, $a = 1$, $b = 1$ and $c = 1$
Now, Determinant, $D = {b^2} - 4ac$
Hence, for $\left( {{z^2} + z + 1} \right) = 0$,
$D = {\left( 1 \right)^2} - 4 \times 1 = 1 - 4 = - 3$
Now, using quadratic formula,
$z = \dfrac{{ - b \pm \sqrt D }}{{2a}}$
Here, $a = 1$, $b = 1$and $c = 1$ and $D = - 3$
$ \Rightarrow z = \dfrac{{ - 1 \pm \sqrt { - 3} }}{2}$
This can be written as:
$ \Rightarrow z = \dfrac{{ - 1 \pm \sqrt 3 i}}{2}$
Therefore, the three cube roots of unity are:
$1$, $\dfrac{{ - 1}}{2} + \dfrac{{\sqrt 3 i}}{2}$ and $\dfrac{{ - 1}}{2} - \dfrac{{\sqrt 3 i}}{2}$
Now, according to the property, the sum of these three cube roots of unity will be equal to 0.
Here, $\omega $ represents the imaginary cube roots.
$ \Rightarrow 1 + \dfrac{{ - 1}}{2} + \dfrac{{\sqrt 3 i}}{2} + \dfrac{{ - 1}}{2} - \dfrac{{\sqrt 3 i}}{2} = 1 - 1 + 0 = 0$
$ \Rightarrow 1 + \omega + {\omega ^2} = 0$
Here, $\omega = \dfrac{{ - 1}}{2} + \dfrac{{\sqrt 3 i}}{2}$
And, ${\omega ^2} = \dfrac{{ - 1}}{2} - \dfrac{{\sqrt 3 i}}{2}$
Therefore,
$1 + \omega + {\omega ^2} = 0$
Hence, proved
Note:
We should know that the cube root of any number is that number which when multiplied two times by itself gives the same number. Hence, the cube root of unity means that when a number is raised to the power 3, it gives the answer 1. The values of cube roots of unity are. Here, one root is a real number and the other two are the imaginary or complex numbers. Hence, we represent the imaginary numbers as $\omega $ and ${\omega ^2}$
Here, we will try to find the three cube roots of unity where one of them will be a real root and the other two will be imaginary roots. When we will add the squares of these three roots together, we will get a 0. Hence, by substituting the imaginary roots by omega and square of omega, we will be able to prove the given identity.
Formula Used:
$\left( {{a^3} - {b^3}} \right) = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)$
Determinant, $D = {b^2} - 4ac$
Quadratic formula, $z = \dfrac{{ - b \pm \sqrt D }}{{2a}}$
Complete step by step solution:
In order to prove the given identity,
Let us assume the cube root of unity or 1 as:
$\sqrt[3]{1} = z$
Cubing both sides, we get
$ \Rightarrow 1 = {z^3}$
Subtracting 1 from both the sides, we get
$ \Rightarrow {z^3} - 1 = 0$
Now, using the formula $\left( {{a^3} - {b^3}} \right) = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)$
$ \Rightarrow \left( {z - 1} \right)\left( {{z^2} + z + 1} \right) = 0$
Therefore, either $\left( {z - 1} \right) = 0$
$ \Rightarrow z = 1$
Or, $\left( {{z^2} + z + 1} \right) = 0$
Comparing with $\left( {a{x^2} + bx + c} \right) = 0$
Here, $a = 1$, $b = 1$ and $c = 1$
Now, Determinant, $D = {b^2} - 4ac$
Hence, for $\left( {{z^2} + z + 1} \right) = 0$,
$D = {\left( 1 \right)^2} - 4 \times 1 = 1 - 4 = - 3$
Now, using quadratic formula,
$z = \dfrac{{ - b \pm \sqrt D }}{{2a}}$
Here, $a = 1$, $b = 1$and $c = 1$ and $D = - 3$
$ \Rightarrow z = \dfrac{{ - 1 \pm \sqrt { - 3} }}{2}$
This can be written as:
$ \Rightarrow z = \dfrac{{ - 1 \pm \sqrt 3 i}}{2}$
Therefore, the three cube roots of unity are:
$1$, $\dfrac{{ - 1}}{2} + \dfrac{{\sqrt 3 i}}{2}$ and $\dfrac{{ - 1}}{2} - \dfrac{{\sqrt 3 i}}{2}$
Now, according to the property, the sum of these three cube roots of unity will be equal to 0.
Here, $\omega $ represents the imaginary cube roots.
$ \Rightarrow 1 + \dfrac{{ - 1}}{2} + \dfrac{{\sqrt 3 i}}{2} + \dfrac{{ - 1}}{2} - \dfrac{{\sqrt 3 i}}{2} = 1 - 1 + 0 = 0$
$ \Rightarrow 1 + \omega + {\omega ^2} = 0$
Here, $\omega = \dfrac{{ - 1}}{2} + \dfrac{{\sqrt 3 i}}{2}$
And, ${\omega ^2} = \dfrac{{ - 1}}{2} - \dfrac{{\sqrt 3 i}}{2}$
Therefore,
$1 + \omega + {\omega ^2} = 0$
Hence, proved
Note:
We should know that the cube root of any number is that number which when multiplied two times by itself gives the same number. Hence, the cube root of unity means that when a number is raised to the power 3, it gives the answer 1. The values of cube roots of unity are. Here, one root is a real number and the other two are the imaginary or complex numbers. Hence, we represent the imaginary numbers as $\omega $ and ${\omega ^2}$
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Which are the Top 10 Largest Countries of the World?
One cusec is equal to how many liters class 8 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
The mountain range which stretches from Gujarat in class 10 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths