Question

Write the following in the form (a+ib):
(2+5i)^3

Answer 1

The given complex number (2 + 5i)³ can be written in the form of (a + ib), as (-142) + i(-65).

Given,

(2 + 5i)³.

To find,

Write the given complex number in the form of (a + ib).

Solution,

It can be seen that here, the given expression is

$(2 + 5i)^3 \hfill ...(1)$

This can be expressed in the form of (a + ib), with the help of the algebraic identity given below.

$(a+b)^{3} =a^3+b^3+3ab(a+b) \hfill ...(2)$

So, using (2), we can write (1) as follows.

$(2+5i)^{3} =(2)^3+(5i)^3+3(2)(5i)(2+5i)$

Simplifying,

$(2+5i)^{3} \\=8+125i^3+30i(2+5i)\\=8-125i+60i+150i^{2} \\=8 - 65i-150\\=-142-65i$

Thus,

$(2+5i)^{3} =-142-65i.$

Or,

$(2+5i)^{3} =(-142)+i(-65),$

which is the required (a + ib) form of $(2+5i)^{3}$, where,

a = -142, and

b = -65.

Therefore, the given complex number (2 + 5i)³ can be written in the form of (a + ib), as (-142) + i(-65).

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