IMPOWER

Returns a complex number raised to a power.

Syntax:

IMPOWER(complexnumber, number)

complexnumber is text representing a complex number, for example as a+bi or a+bj. number is a number.

IMPOWER returns complexnumber raised to the power of number. The result is a complex number presented as text.

If complexnumber is $a + bi = r (cos φ + i s in φ)$ and number is n, IMPOWER(complexnumber, number) returns $(a + bi)^{n} = r^{n} cos (n φ) + i r^{n} s in (n φ)$ .

Example:

IMPOWER("1+2i", 2)

returns (1+2i)2 = -3+4i as text.

Application:

An application of using the IMPOWER function is in electrical circuit analysis, specifically in alternating current (AC) circuits. In AC circuits, components like resistors, capacitors, and inductors have an impedance, which is often represented as a complex number. The impedance describes the circuit's opposition to the flow of alternating current.

Consider a simple AC circuit where you need to calculate the total impedance of a series of components. Let's say you have a capacitor whose impedance is given by the complex number ZC=0−3i ohms and you need to find the impedance of a circuit where this capacitor is part of a more complex structure that requires raising its impedance to a specific power.

Here's an example using a table format, as you requested:

In this example, we have a complex number representing the impedance of a component and we need to calculate its value when raised to different powers.

	Complex Number	Power	IMPOWER Function Formula	Result
	A	B	C	D
1	2+3i	2	IMPOWER(A1, B1)	−5+12i
2	2+3i	3	IMPOWER(A2, B2)	−46+9i
3	2-5j	0.5	IMPOWER(A3, B3)	2.203−1.135j
4	1+i	4	IMPOWER(A4, B4)	−4+0i

Explanation:

Row 1: The complex number 2+3i is squared. The IMPOWER function calculates (2+3i)2=4+12i+9i2=4+12i−9=−5+12i.
Row 2: The complex number 2+3i is cubed. The IMPOWER function performs the calculation (2+3i)3=(2+3i)(2+3i)2=(2+3i)(−5+12i)=−10+24i−15i+36i2=−10+9i−36=−46+9i.
Row 3: The complex number 2−5j is raised to the power of 0.5, which is the same as taking the square root. The IMPOWER function handles this fractional power and provides the complex square root.
Row 4: The complex number 1+i is raised to the fourth power. The IMPOWER function calculates (1+i)4=((1+i)2)2=(1+2i+i2)2=(1+2i−1)2=(2i)2=4i2=−4.