In Exercises 1–3, perform the indicated operations and write the result in standard form. ___ ___ 2√−49 + 3√−64

Verified step by step guidance

1. 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 \(i^2 = -1\).
2. Rewrite each term by separating the negative sign inside the square root: \(2\sqrt{-49} = 2\sqrt{49} \times i\) and \(3\sqrt{-64} = 3\sqrt{64} \times i\).
3. Calculate the square roots of the positive numbers: \(\sqrt{49} = 7\) and \(\sqrt{64} = 8\).
4. Substitute these values back into the expression: \(2 \times 7 \times i + 3 \times 8 \times i\).
5. Combine like terms by factoring out \(i\): \((2 \times 7 + 3 \times 8) i\), which simplifies to \((14 + 24) i\). This is the expression in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part.

Key Concepts

Imaginary Numbers and the Imaginary Unit i

Imaginary numbers arise when taking the square root of negative numbers, which is not defined in the real number system. The imaginary unit i is defined as √(-1), allowing us to express roots of negative numbers as multiples of i, such as √(-49) = 7i.

Simplifying Radicals Involving Negative Numbers

To simplify radicals with negative radicands, factor out the negative sign as i² or i, then simplify the positive part. For example, √(-64) = √(64) × √(-1) = 8i. This process helps convert complex radicals into a standard form involving real and imaginary parts.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers and i is the imaginary unit. After performing operations on imaginary numbers, results should be expressed in this form to clearly separate the real and imaginary components.