Question

express the given complex number in the form a + ib: {1/3+3i}^3​​

Answer 1

Answer:

The complex number is the form of a + ib is

$\displaystyle{\boxed{\red{\sf\:-\:\dfrac{242}{27}\:+\:(\:-\:26\:)\:i\:}}}$

Step-by-step-explanation:

The given complex number is

$\displaystyle{\sf\:\left(\:\dfrac{1}{3}\:+\:3i\:\right)^3}$

We have to express this complex number in the form of a + ib.

Let the given complex number be A.

$\displaystyle{\sf\:A\:=\:\left(\:\dfrac{1}{3}\:+\:3i\:\right)^3}$

We know that,

$\displaystyle{\boxed{\pink{\sf\:(\:a\:+\:b\:)^3\:=\:a^3\:+\:3a^2b\:+\:3ab^2\:+\:b^3\:}}}$

$\displaystyle{\implies\sf\:A\:=\:\left(\:\dfrac{1}{3}\:\right)^3\:+\:3\:\times\:\left(\:\dfrac{1}{3}\:\right)^2\:\times\:3i\:+\:3\:\times\:\dfrac{1}{3}\:\times\:(\:3i\:)^2\:+\:(\:3i\:)^3}$

$\displaystyle{\implies\sf\:A\:=\:\dfrac{1}{27}\:+\:3\:\times\:\dfrac{1}{3}\:\times\:\dfrac{1}{3}\:\times\:3\:i\:+\:9\:i^2\:+\:27\:i^3}$

We know that,

• i² = - 1
• i³ = - i

$\displaystyle{\implies\sf\:A\:=\:\dfrac{1}{27}\:+\:i\:+\:9\:\times\:(\:-\:1\:)\:+\:27\:\times\:(\:-\:i\:)}$

$\displaystyle{\implies\sf\:A\:=\:\dfrac{1}{27}\:+\:i\:-\:9\:-\:27i}$

$\displaystyle{\implies\sf\:A\:=\:\dfrac{1}{27}\:-\:9\:+\:i\:-\:27i}$

$\displaystyle{\implies\sf\:A\:=\:\dfrac{1\:-\:9\:\times\:27}{27}\:-\:26i}$

$\displaystyle{\implies\sf\:A\:=\:\dfrac{1\:-\:243}{27}\:-\:26i}$

$\displaystyle{\implies\:\underline{\boxed{\red{\sf\:A\:=\:-\:\dfrac{242}{27}\:-\:26i\:}}}}$

∴ The complex number is the form of a + ib is

$\displaystyle{\boxed{\red{\sf\:-\:\dfrac{242}{27}\:+\:(\:-\:26\:)\:i\:}}}$