How do you simplify #i^3(2i^6-4i^21)#?

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Answer 1 · 2016-02-09T16:47:52

#i^3(2i^6 - 4 i^21) = 2i -4#

You should use the power laws

[1] #" "a^m * a^n = a^(m+n)#

[2] #" "(a^m)^n = a^(m*n)#

Also, remember that

[3] #" "i^2 = - 1#

and with the help of [3], you can compute

[4] #" "i^4 = i^2 * i^2 = (-1) * (-1) = 1#

Thus, you can simplify as follows:

#i^3(2i^6 - 4 i^21) = i^3 * 2 i^6 - i^3 * 4 i^21#

#stackrel("[1] ")(=) 2i^(3+6) - 4 i^(3+21)#

#= 2i^9 - 4 i^24#

Now, let's evaluate #i^9# and #i^24#:

#i^9 = i^(4+4+1) = i^4 * i^4 * i^1 stackrel("[4] ")(=) 1 * 1 * i = i#

#i^24 = i^(6*4) = i^(6*4) stackrel("[2] ")(=) (i^4)^6= 1^6 = 1#

Thus, your result is

#i^3(2i^6 - 4 i^21) = 2i^9 - 4 i^24 = 2i - 4 #

Hope that this helped!