QuestionFind the value of $i^{120}$ .

Studdy Solution

STEP 1

Assumptions1. We are dealing with complex numbers, where $i$ is the imaginary unit with the property that $i^ = -1$ .

STEP 2

We know that the powers of $i$ repeat every4 cycles, i.e., $i^1 = i$ , $i^2 = -1$ , $i^ = -i$ , $i^4 =1$ , and then $i^5 = i$ , $i^6 = -1$ , and so on.

STEP 3

Therefore, we can simplify $i^{120}$ by finding the remainder when120 is divided by, because the powers of $i$ repeat every cycles.
$i^{120} = i^{ \times30}$

STEP 4

Since $i^4 =1$ , we can simplify $i^{4 \times30}$ to $1^{30}$ .
$i^{4 \times30} =1^{30}$

STEP 5

Finally, any number to the power of30 is itself, so $1^{30} =1$ .
$1^{30} =1$ So, $i^{120} =1$ .

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QuestionFind the value of $i^{120}$ .

Studdy Solution

STEP 1

Assumptions1. We are dealing with complex numbers, where $i$ is the imaginary unit with the property that $i^ = -1$ .

STEP 2

We know that the powers of $i$ repeat every4 cycles, i.e., $i^1 = i$ , $i^2 = -1$ , $i^ = -i$ , $i^4 =1$ , and then $i^5 = i$ , $i^6 = -1$ , and so on.

STEP 3

Therefore, we can simplify $i^{120}$ by finding the remainder when120 is divided by, because the powers of $i$ repeat every cycles.
$i^{120} = i^{ \times30}$

STEP 4

Since $i^4 =1$ , we can simplify $i^{4 \times30}$ to $1^{30}$ .
$i^{4 \times30} =1^{30}$

STEP 5

Finally, any number to the power of30 is itself, so $1^{30} =1$ .
$1^{30} =1$ So, $i^{120} =1$ .

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QuestionFind the value of i120i^{120}i120.

Studdy Solution

STEP 1

Assumptions1. We are dealing with complex numbers, where iii is the imaginary unit with the property that i=−1i^ = -1i=−1.

STEP 2

We know that the powers of iii repeat every4 cycles, i.e., i1=ii^1 = ii1=i, i2=−1i^2 = -1i2=−1, i=−ii^ = -ii=−i, i4=1i^4 =1i4=1, and then i5=ii^5 = ii5=i, i6=−1i^6 = -1i6=−1, and so on.

STEP 3

Therefore, we can simplify i120i^{120}i120 by finding the remainder when120 is divided by, because the powers of iii repeat every cycles.i120=i×30i^{120} = i^{ \times30}i120=i×30

STEP 4

Since i4=1i^4 =1i4=1, we can simplify i4×30i^{4 \times30}i4×30 to 1301^{30}130.i4×30=130i^{4 \times30} =1^{30}i4×30=130

STEP 5

Finally, any number to the power of30 is itself, so 130=11^{30} =1130=1.130=11^{30} =1130=1So, i120=1i^{120} =1i120=1.

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QuestionFind the value of $i^{120}$ .

Assumptions1. We are dealing with complex numbers, where $i$ is the imaginary unit with the property that $i^ = -1$ .

We know that the powers of $i$ repeat every4 cycles, i.e., $i^1 = i$ , $i^2 = -1$ , $i^ = -i$ , $i^4 =1$ , and then $i^5 = i$ , $i^6 = -1$ , and so on.

Therefore, we can simplify $i^{120}$ by finding the remainder when120 is divided by, because the powers of $i$ repeat every cycles.
$i^{120} = i^{ \times30}$

Since $i^4 =1$ , we can simplify $i^{4 \times30}$ to $1^{30}$ .
$i^{4 \times30} =1^{30}$

Finally, any number to the power of30 is itself, so $1^{30} =1$ .
$1^{30} =1$ So, $i^{120} =1$ .