Answer :
To write the expression [tex]\(\sqrt{-18} \sqrt{-28}\)[/tex] in standard form, let's go through the steps:
1. Understanding the Components:
- [tex]\(\sqrt{-18}\)[/tex] and [tex]\(\sqrt{-28}\)[/tex] are both square roots of negative numbers. We will handle the negative sign by representing each square root using the imaginary unit [tex]\(i\)[/tex], where [tex]\(i^2 = -1\)[/tex].
- [tex]\(\sqrt{-18} = \sqrt{18} \cdot i\)[/tex] and [tex]\(\sqrt{-28} = \sqrt{28} \cdot i\)[/tex].
2. Calculate Square Roots:
- [tex]\(\sqrt{18}\)[/tex] can be simplified. Since [tex]\(18 = 9 \times 2\)[/tex], we have [tex]\(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\)[/tex].
- [tex]\(\sqrt{28}\)[/tex] can also be simplified. Since [tex]\(28 = 4 \times 7\)[/tex], we have [tex]\(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}\)[/tex].
3. Substitute and Simplify:
- Substitute the simplified forms back into the expression: [tex]\(\left(3\sqrt{2} \cdot i\right) \left(2\sqrt{7} \cdot i\right)\)[/tex].
4. Multiply the Expressions:
- Multiply the constants and the square roots: [tex]\(3 \cdot 2 = 6\)[/tex] and [tex]\(\sqrt{2} \cdot \sqrt{7} = \sqrt{14}\)[/tex].
- Also multiply the imaginary units: [tex]\(i \cdot i = i^2 = -1\)[/tex].
5. Combine Everything:
- The combined expression is: [tex]\(6 \cdot \sqrt{14} \cdot -1 = -6\sqrt{14}\)[/tex].
Thus, the standard form of [tex]\(\sqrt{-18} \sqrt{-28}\)[/tex] is [tex]\(-6\sqrt{14}\)[/tex].
1. Understanding the Components:
- [tex]\(\sqrt{-18}\)[/tex] and [tex]\(\sqrt{-28}\)[/tex] are both square roots of negative numbers. We will handle the negative sign by representing each square root using the imaginary unit [tex]\(i\)[/tex], where [tex]\(i^2 = -1\)[/tex].
- [tex]\(\sqrt{-18} = \sqrt{18} \cdot i\)[/tex] and [tex]\(\sqrt{-28} = \sqrt{28} \cdot i\)[/tex].
2. Calculate Square Roots:
- [tex]\(\sqrt{18}\)[/tex] can be simplified. Since [tex]\(18 = 9 \times 2\)[/tex], we have [tex]\(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\)[/tex].
- [tex]\(\sqrt{28}\)[/tex] can also be simplified. Since [tex]\(28 = 4 \times 7\)[/tex], we have [tex]\(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}\)[/tex].
3. Substitute and Simplify:
- Substitute the simplified forms back into the expression: [tex]\(\left(3\sqrt{2} \cdot i\right) \left(2\sqrt{7} \cdot i\right)\)[/tex].
4. Multiply the Expressions:
- Multiply the constants and the square roots: [tex]\(3 \cdot 2 = 6\)[/tex] and [tex]\(\sqrt{2} \cdot \sqrt{7} = \sqrt{14}\)[/tex].
- Also multiply the imaginary units: [tex]\(i \cdot i = i^2 = -1\)[/tex].
5. Combine Everything:
- The combined expression is: [tex]\(6 \cdot \sqrt{14} \cdot -1 = -6\sqrt{14}\)[/tex].
Thus, the standard form of [tex]\(\sqrt{-18} \sqrt{-28}\)[/tex] is [tex]\(-6\sqrt{14}\)[/tex].
