Video Transcript
Given that 𝑛 is an integer,
simplify 𝑖 to the power of 16𝑛 minus 35.
Remember the cycle that helps us
remember the identities for various powers of 𝑖 is as shown. So we can do one of two things. Our first method is to use the laws
of exponents to essentially unsimplify our expression a little. We know that 𝑥 to the power of 𝑎
times 𝑥 to the power of 𝑏 is the same as 𝑥 to the power of 𝑎 plus 𝑏. So we can reverse this and say that
𝑖 to the power of 16𝑛 minus 35 is equal to 𝑖 to the power of 16𝑛 times 𝑖 to the
power of negative 35.
𝑖 to the power of 16𝑛 can
actually be further written as 𝑖 to the power of four to the power of four 𝑛. This corresponds to the part of our
cycle 𝑖 to the power of four 𝑛. So we can see that 𝑖 to the power
of 16𝑛 can be written as one. And what about 𝑖 to the power of
negative 35? This one is a little more
complicated. We’re going to write negative 35 in
the form four 𝑎 plus 𝑏, where 𝑏 can take the values zero, one, two, or three to
correspond with the values in our cycle. It’s the same as four times
negative nine plus one.
Remember four times negative nine
is negative 36 and adding one gets us negative 35. And we chose negative nine instead
of negative eight as we needed 𝑏 to be zero, one, two, or three. And we certainly don’t want it to
be a negative value. So 𝑖 to the power of negative 35
will have the same result as 𝑖 to the power of four 𝑛 plus one in our cycle;
that’s 𝑖. So 𝑖 to the power of 16𝑛 minus 35
is one multiplied by 𝑖 which is 𝑖.
Let’s have a look at the
alternative method. Here we would have jumped straight
into writing the power — that’s 16 𝑛 minus 35 — in the form four 𝑎 plus 𝑏, where
𝑏 again is zero, one, two, or three. We can write 16𝑛 as four times
four 𝑛 and negative 35 as four times negative nine plus one. We can factor this expression and
we see that 16 𝑛 minus 35 is the same as four multiplied by four 𝑛 minus nine plus
one. So we can see that once again 𝑖 to
the power of 16𝑛 minus 35 will have the same result as 𝑖 to the power of four 𝑛
plus one in our cycle; that’s 𝑖.
