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

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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: \(2\sqrt{-49} = 2\sqrt{49} \times i\) and \(3\sqrt{-64} = 3\sqrt{64} \times i\).

Calculate the square roots of the positive numbers: \(\sqrt{49} = 7\) and \(\sqrt{64} = 8\).

Substitute these values back into the expression: \(2 \times 7 \times i + 3 \times 8 \times i\).

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.

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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.

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