In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. _ _ 5√(−16) + 3√(−81)

검증된 단계별 안내

Recognize that the expressions involve square roots of negative numbers, which means we are dealing with imaginary numbers. Recall that $\sqrt{-a} = \sqrt{a} \times i$, where $i$ is the imaginary unit with the property $i^2 = -1$.

Rewrite each term by separating the negative sign inside the square root: $5\sqrt{-16} = 5 \times \sqrt{16} \times i$ and $3\sqrt{-81} = 3 \times \sqrt{81} \times i$.

Calculate the square roots of the positive numbers: $\sqrt{16} = 4$ and $\sqrt{81} = 9$, so the terms become $5 \times 4 \times i$ and $3 \times 9 \times i$ respectively.

Multiply the coefficients: $5 \times 4 = 20$ and $3 \times 9 = 27$, so the expression is now $20i + 27i$.

Combine like terms (both are imaginary terms) by adding the coefficients: $20i + 27i = (20 + 27)i$, which simplifies to \$47i$. This is the expression in standard form, where the real part is 0 and the imaginary part is $47i$.

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 include a real part and an imaginary part, where the imaginary unit 'i' is defined as √−1. When dealing with square roots of negative numbers, express them in terms of 'i' to simplify and perform operations.

Simplifying Square Roots of Negative Numbers

To simplify √−a, rewrite it as √a × √−1, which equals √a × i. For example, √−16 becomes 4i. This allows you to convert roots of negative numbers into imaginary numbers for further calculation.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary part. After performing operations, express the result in this form for clarity and consistency.

In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. ___ ___ 5√(−16) + 3√(−81)