Perform the indicated operations and write the result in standard form. −8(−3−−5)\sqrt{-8} \left( \sqrt{-3} - \sqrt{-5} \right) −8(−3−−5)

Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 49

2장, 문제 49

Perform the indicated operations and write the result in standard form. $$\sqrt{-8}$ $\left$( $\sqrt{-3}$ - $\sqrt{-5}$ $\right$)$

검증된 단계별 안내

Recognize that the expression involves square roots of negative numbers, which means we will be working with imaginary numbers. Recall that $ \sqrt{-1} = i $.

Rewrite the square roots of negative numbers using $ i $: $ \sqrt{-8} = \sqrt{8} \cdot i $ and $ \sqrt{-3} = \sqrt{3} \cdot i $.

Simplify $ \sqrt{8} $ as $ 2\sqrt{2} $, so $ \sqrt{-8} = 2\sqrt{2}i $.

Substitute the expressions back into the original problem: $ (2\sqrt{2}i)(\sqrt{3}i - \sqrt{5}) $.

Distribute $ 2\sqrt{2}i $ across the terms inside the parentheses: $ 2\sqrt{2}i \cdot \sqrt{3}i - 2\sqrt{2}i \cdot \sqrt{5} $.

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주요 개념

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

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as √-1. Understanding complex numbers is essential for performing operations involving square roots of negative numbers, as they allow us to extend the real number system.

Square Roots of Negative Numbers

The square root of a negative number is not defined within the real number system, but it can be expressed using imaginary numbers. For example, √-8 can be simplified to 2√2i, where 'i' represents the imaginary unit. This concept is crucial for solving problems that involve square roots of negative values, as it allows for the manipulation of these expressions in algebraic operations.

Standard Form of Complex Numbers

The standard form of a complex number is typically written as a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, such as addition or multiplication, it is important to express the final result in this standard form for clarity and consistency. This involves combining like terms and ensuring that the imaginary unit 'i' is properly accounted for in the final expression.

Perform the indicated operations and write the result in standard form. −8(−3−−5)\(\sqrt{-8}\) \(\left\)( \(\sqrt{-3}\) - \(\sqrt{-5}\) \(\right\)) −8(−3−−5)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Complex Numbers

Square Roots of Negative Numbers

Standard Form of Complex Numbers

Perform the indicated operations and write the result in standard form. $\(\sqrt{-8}\) \(\left\)( \(\sqrt{-3}\) - \(\sqrt{-5}\) \(\right\))$