<< Chapter < Page Chapter >> Page >

Simplifying powers of i

The powers of i are cyclic. Let’s look at what happens when we raise i to increasing powers.

i 1 = i i 2 = −1 i 3 = i 2 i = −1 i = i i 4 = i 3 i = i i = i 2 = ( −1 ) = 1 i 5 = i 4 i = 1 i = i

We can see that when we get to the fifth power of i , it is equal to the first power. As we continue to multiply i by increasing powers, we will see a cycle of four. Let’s examine the next four powers of i .

i 6 = i 5 i = i i = i 2 = −1 i 7 = i 6 i = i 2 i = i 3 = i i 8 = i 7 i = i 3 i = i 4 = 1 i 9 = i 8 i = i 4 i = i 5 = i

The cycle is repeated continuously: i , −1 , i , 1 , every four powers.

Simplifying powers of i

Evaluate: i 35 .

Since i 4 = 1 , we can simplify the problem by factoring out as many factors of i 4 as possible. To do so, first determine how many times 4 goes into 35: 35 = 4 8 + 3.

i 35 = i 4 8 + 3 = i 4 8 i 3 = ( i 4 ) 8 i 3 = 1 8 i 3 = i 3 = i
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Can we write i 35 in other helpful ways?

As we saw in [link] , we reduced i 35 to i 3 by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of i 35 may be more useful. [link] shows some other possible factorizations.

Factorization of i 35 i 34 i i 33 i 2 i 31 i 4 i 19 i 16
Reduced form ( i 2 ) 17 i i 33 ( −1 ) i 31 1 i 19 ( i 4 ) 4
Simplified form ( −1 ) 17 i i 33 i 31 i 19

Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.

Access these online resources for additional instruction and practice with complex numbers.

Key concepts

  • The square root of any negative number can be written as a multiple of i . See [link] .
  • To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See [link] .
  • Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See [link] .
  • Complex numbers can be multiplied and divided.
    • To multiply complex numbers, distribute just as with polynomials. See [link] and [link] .
    • To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See [link] and [link] .
  • The powers of i are cyclic, repeating every fourth one. See [link] .

Section exercises

Verbal

Explain how to add complex numbers.

Add the real parts together and the imaginary parts together.

Got questions? Get instant answers now!

What is the basic principle in multiplication of complex numbers?

Got questions? Get instant answers now!

Give an example to show that the product of two imaginary numbers is not always imaginary.

Possible answer: i times i equals 1, which is not imaginary.

Got questions? Get instant answers now!

What is a characteristic of the plot of a real number in the complex plane?

Got questions? Get instant answers now!

Algebraic

For the following exercises, evaluate the algebraic expressions.

If y = x 2 + x 4 , evaluate y given x = 2 i .

−8 + 2 i

Got questions? Get instant answers now!

If y = x 3 2 , evaluate y given x = i .

Got questions? Get instant answers now!

If y = x 2 + 3 x + 5 , evaluate y given x = 2 + i .

14 + 7 i

Got questions? Get instant answers now!

If y = 2 x 2 + x 3 , evaluate y given x = 2 3 i .

Got questions? Get instant answers now!

If y = x + 1 2 x , evaluate y given x = 5 i .

23 29 + 15 29 i

Got questions? Get instant answers now!

If y = 1 + 2 x x + 3 , evaluate y given x = 4 i .

Got questions? Get instant answers now!

Graphical

For the following exercises, plot the complex numbers on the complex plane.

Numeric

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

( 3 + 2 i ) + ( 5 3 i )

8 i

Got questions? Get instant answers now!

( −2 4 i ) + ( 1 + 6 i )

Got questions? Get instant answers now!

( −5 + 3 i ) ( 6 i )

−11 + 4 i

Got questions? Get instant answers now!

( 2 3 i ) ( 3 + 2 i )

Got questions? Get instant answers now!

( −4 + 4 i ) ( −6 + 9 i )

2 −5 i

Got questions? Get instant answers now!

( 5 2 i ) ( 3 i )

6 + 15 i

Got questions? Get instant answers now!

( −2 + 4 i ) ( 8 )

−16 + 32 i

Got questions? Get instant answers now!

( −1 + 2 i ) ( −2 + 3 i )

−4 −7 i

Got questions? Get instant answers now!

( 4 2 i ) ( 4 + 2 i )

Got questions? Get instant answers now!

( 3 + 4 i ) ( 3 4 i )

25

Got questions? Get instant answers now!

3 + 4 i 2 i

2 5 + 11 5 i

Got questions? Get instant answers now!

Technology

For the following exercises, use a calculator to help answer the questions.

Evaluate ( 1 + i ) k for k = 4 , 8 , and 12. Predict the value if k = 16.

Got questions? Get instant answers now!

Evaluate ( 1 i ) k for k = 2 , 6 , and 10. Predict the value if k = 14.

128i

Got questions? Get instant answers now!

Evaluate ( l + i ) k ( l i ) k for k = 4 , 8 , and 12. Predict the value for k = 16.

Got questions? Get instant answers now!

Show that a solution of x 6 + 1 = 0 is 3 2 + 1 2 i .

( 3 2 + 1 2 i ) 6 = −1

Got questions? Get instant answers now!

Show that a solution of x 8 −1 = 0 is 2 2 + 2 2 i .

Got questions? Get instant answers now!

Extensions

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

( 2 + i ) ( 4 2 i ) ( 1 + i )

5 −5 i

Got questions? Get instant answers now!

( 1 + 3 i ) ( 2 4 i ) ( 1 + 2 i )

Got questions? Get instant answers now!

( 3 + i ) 2 ( 1 + 2 i ) 2

−2 i

Got questions? Get instant answers now!

3 + 2 i 2 + i + ( 4 + 3 i )

Got questions? Get instant answers now!

4 + i i + 3 4 i 1 i

9 2 9 2 i

Got questions? Get instant answers now!

3 + 2 i 1 + 2 i 2 3 i 3 + i

Got questions? Get instant answers now!

Questions & Answers

More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
modul questions trigonometry
Thamarai Reply
(1+cosA)(1-cosA)=sin^2A
BINCY
differentiate f(t)=1/4t to the power 4 +8
Jessica Reply
I need trigonometry,polynomial
duru Reply
ok
Augustine
Why is 7 on top
Bertha Reply
simplify cot x / csc x
Catherine Reply
👉🌹Solve🌻 Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
what is the period of cos?
SIYAMTHEMBA Reply
your question might not seem clear as you asked. ask well to get perfect answers put your question on a table I'm willing to help you Mr Siyamthemba
Patrick
simplify: cot x/csc x
Catherine
sorry i didnt realize you were actually asking someone else to put their question on here. i thought this was where i was supposed to.
Catherine
some to dereve formula for bulky density
kurash
Solve Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
questions
Thamarai
ok
AjA
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask