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 expression involves square roots of negative numbers, which means we are dealing with imaginary numbers.
2. Rewrite the square roots of negative numbers using the imaginary unit $i$, where $i=\sqrt{-1}$.
3. Express $\sqrt{-49}$ as $\sqrt{49}\cdot \sqrt{-1}=7i$ and $\sqrt{-64}$ as $\sqrt{64}\cdot \sqrt{-1}=8i$.
4. Substitute these expressions back into the original problem: $2\cdot 7i+3\cdot 8i$.
5. Simplify the expression by performing the multiplication and addition: $14i+24i$.

Key Concepts

Imaginary Numbers

Imaginary numbers are defined as multiples of the imaginary unit 'i', where i is the square root of -1. They arise when taking the square root of negative numbers, which is not possible within the realm of real numbers. For example, √-49 can be expressed as 7i, since √49 = 7 and the negative sign introduces the imaginary unit.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' is the real part and 'b' is the imaginary part. In the context of the given problem, after performing the indicated operations, the result should be simplified to this form. This allows for easier interpretation and manipulation of complex numbers in mathematical operations.

Operations with Complex Numbers

Operations with complex numbers include addition, subtraction, multiplication, and division. When adding or subtracting complex numbers, you combine the real parts and the imaginary parts separately. For instance, when adding 2√-49 and 3√-64, you first convert them to their imaginary forms and then combine like terms to arrive at the final result in standard form.