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!

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