IDNLearn.com provides a collaborative platform for sharing and gaining knowledge. Whether it's a simple query or a complex problem, our community has the answers you need.
Sagot :
To simplify the expression [tex]\(i^{37}\)[/tex], we can use the properties of the imaginary unit [tex]\(i\)[/tex] and its powers. Recall the cyclic nature of the powers of [tex]\(i\)[/tex]:
[tex]\[ \begin{aligned} i^1 &= i, \\ i^2 &= -1, \\ i^3 &= -i, \\ i^4 &= 1. \end{aligned} \][/tex]
Notice that [tex]\(i^4 = 1\)[/tex], which means any power of [tex]\(i\)[/tex] that is a multiple of [tex]\(4\)[/tex] will be equal to [tex]\(1\)[/tex]. Because of this, we can reduce the exponent [tex]\(37\)[/tex] modulo [tex]\(4\)[/tex] to simplify the expression.
Let's perform the steps:
1. Reduce the exponent modulo 4:
[tex]\[ 37 \mod 4 = 1. \][/tex]
This tells us that:
[tex]\[ i^{37} \equiv i^1 \pmod{4}. \][/tex]
Thus:
[tex]\[ i^{37} = i^1. \][/tex]
2. Identify the equivalent power of [tex]\(i\)[/tex]:
Based on the calculations above:
[tex]\[ i^{37} = i. \][/tex]
So finally, the simplified expression for [tex]\(i^{37}\)[/tex] is:
[tex]\[ i^{37} = i. \][/tex]
[tex]\(\boxed{i}\)[/tex]
Feel free to ask if you have any further questions regarding simplifying expressions that involve the imaginary unit [tex]\(i\)[/tex]!
[tex]\[ \begin{aligned} i^1 &= i, \\ i^2 &= -1, \\ i^3 &= -i, \\ i^4 &= 1. \end{aligned} \][/tex]
Notice that [tex]\(i^4 = 1\)[/tex], which means any power of [tex]\(i\)[/tex] that is a multiple of [tex]\(4\)[/tex] will be equal to [tex]\(1\)[/tex]. Because of this, we can reduce the exponent [tex]\(37\)[/tex] modulo [tex]\(4\)[/tex] to simplify the expression.
Let's perform the steps:
1. Reduce the exponent modulo 4:
[tex]\[ 37 \mod 4 = 1. \][/tex]
This tells us that:
[tex]\[ i^{37} \equiv i^1 \pmod{4}. \][/tex]
Thus:
[tex]\[ i^{37} = i^1. \][/tex]
2. Identify the equivalent power of [tex]\(i\)[/tex]:
Based on the calculations above:
[tex]\[ i^{37} = i. \][/tex]
So finally, the simplified expression for [tex]\(i^{37}\)[/tex] is:
[tex]\[ i^{37} = i. \][/tex]
[tex]\(\boxed{i}\)[/tex]
Feel free to ask if you have any further questions regarding simplifying expressions that involve the imaginary unit [tex]\(i\)[/tex]!
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.