How do you simplify #i44 + i150 - i74 - i109 + i61#?

Answer 1

1

We have, #i^1=i# #i^2= -1# #i^3 = i^2 xx i = -i# #i^4 = i^2 xx i^2 =(-1)^2 xx (-1)^2 = 1# #i^5 = i^4 xx i = 1 xx i =i# #i^6 = i^4 * i^2 = 1 * -1 = -1# #i^7 = i^4 * i^3 = 1 * -i = -i # #i^8 = i^4 * i^4 = 1 * 1 = 1#
Now, #i^44 = (i^4) ^11 = (1)^11 = 1# #i^150 = (i^4)^37 * i^2 = i^2 = -1# #i^74 = (i^4)^18 * i^2 = i^2 = -1# #i^109 = (i^4)^27 * i^1 = i # #i^61 = (i^4)^15 * i^1 = i #
Finally, #i^44 + i^150 - i^74 - i^109 + i^61# #=(1) + (-1) - (-1) - (i) + (i)# #=1 - 1 + 1 - i + i# #= 1 #
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Answer 2

To simplify the expression (i^{44} + i^{150} - i^{74} - i^{109} + i^{61}), we need to consider the powers of (i):

(i^{44}) is equivalent to (i^{4 \times 11}), and since (i^4) equals 1, (i^{44} = 1).

Similarly, (i^{150}) is equivalent to (i^{4 \times 37}), and (i^{4}) equals 1, so (i^{150} = 1).

For (i^{74}), it's equivalent to (i^{4 \times 18 + 2}), and (i^{4 \times 18}) equals 1, while (i^2) equals -1, so (i^{74} = -1).

For (i^{109}), it's equivalent to (i^{4 \times 27 + 1}), and (i^{4 \times 27}) equals 1, while (i^1) equals (i), so (i^{109} = i).

Lastly, (i^{61}) is equivalent to (i^{4 \times 15 + 1}), and (i^{4 \times 15}) equals 1, while (i^1) equals (i), so (i^{61} = i).

Now, substituting these values into the expression:

[1 + 1 - (-1) - i + i] [= 1 + 1 + 1 - i + i] [= 3]

Therefore, the simplified expression is 3.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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