Work each problem. Show that √2/2 + √2/2 i is a square root of i.

검증된 단계별 안내

Express the given complex number as \(z = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\).

Recall that to show \(z\) is a square root of \(i\), we need to verify that \(z^2 = i\).

Calculate \(z^2\) by using the formula for squaring a complex number: \((a + bi)^2 = a^2 + 2abi + (bi)^2\) where \(a = \frac{\sqrt{2}}{2}\) and \(b = \frac{\sqrt{2}}{2}\).

Substitute the values of \(a\) and \(b\) into the formula and simplify each term carefully, remembering that \(i^2 = -1\).

After simplification, compare the result of \(z^2\) with \(i\) to confirm whether they are equal, thus proving \(z\) is a square root of \(i\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Complex Numbers and Imaginary Unit

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. Understanding how to work with i is essential for manipulating and interpreting expressions involving complex numbers.

Square Roots of Complex Numbers

Finding the square root of a complex number involves determining a complex number which, when squared, equals the original number. This requires knowledge of how to square complex numbers and equate real and imaginary parts to verify the root.

Multiplication of Complex Numbers in Standard Form

Multiplying complex numbers in the form a + bi involves using the distributive property and the fact that i² = -1. This process helps verify if a given complex number squared equals another complex number, such as checking if (√2/2 + √2/2 i)² equals i.