<< 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

sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak Reply
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