# How do you simplify #i^(-43)+i^(-32)# ?

So in general we can write:

Now:

So:

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To simplify ( i^{-43} + i^{-32} ), we first need to recall the properties of the imaginary unit, ( i ). The imaginary unit ( i ) is defined as ( \sqrt{-1} ), and it follows the pattern: [ i^1 = i, \ i^2 = -1, \ i^3 = -i, \ i^4 = 1, \ i^5 = i, \ \text{and so on.} ]

Since the powers of ( i ) repeat every four terms, we can simplify the given expression by finding the remainder when each exponent is divided by 4: [ i^{-43} = i^{(-4 \times 10) - 3} = i^{-3}, ] [ i^{-32} = i^{(-4 \times 8) + 0} = i^0. ]

Now, using the pattern of ( i ) mentioned earlier: [ i^{-3} = -i, ] [ i^0 = 1. ]

Finally, we add the simplified terms: [ i^{-43} + i^{-32} = -i + 1. ]

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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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