Certainly! Let's simplify the sum of the square roots of negative numbers step-by-step.
Given terms are:
[tex]\[ \sqrt{-36} + \sqrt{-49} + \sqrt{-64} \][/tex]
First, we use the property of square roots with negative numbers, [tex]\(\sqrt{-a} = i\sqrt{a}\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit, defined by [tex]\(i^2 = -1\)[/tex].
1. Simplifying each term:
- For [tex]\(\sqrt{-36}\)[/tex]:
[tex]\[
\sqrt{-36} = \sqrt{-1 \times 36} = \sqrt{-1} \times \sqrt{36} = i \times 6 = 6i
\][/tex]
- For [tex]\(\sqrt{-49}\)[/tex]:
[tex]\[
\sqrt{-49} = \sqrt{-1 \times 49} = \sqrt{-1} \times \sqrt{49} = i \times 7 = 7i
\][/tex]
- For [tex]\(\sqrt{-64}\)[/tex]:
[tex]\[
\sqrt{-64} = \sqrt{-1 \times 64} = \sqrt{-1} \times \sqrt{64} = i \times 8 = 8i
\][/tex]
2. Adding the simplified terms:
[tex]\[
\sqrt{-36} + \sqrt{-49} + \sqrt{-64} = 6i + 7i + 8i
\][/tex]
3. Combine like terms:
[tex]\[
6i + 7i + 8i = (6 + 7 + 8)i = 21i
\][/tex]
Therefore, the simplified form of the given expression [tex]\(\sqrt{-36} + \sqrt{-49} + \sqrt{-64}\)[/tex] is:
[tex]\[ 21i \][/tex]
