To multiply the expressions [tex]\((2, \sqrt{-25})(4-\sqrt{-100})\)[/tex], we need to simplify each term first and then perform the multiplication step-by-step.
### Step 1: Simplify the square roots of negative numbers
1. [tex]\(\sqrt{-25}\)[/tex]:
- We know that [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
- Therefore, [tex]\(\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i\)[/tex].
2. [tex]\(\sqrt{-100}\)[/tex]:
- Similarly, [tex]\(\sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} = 10i\)[/tex].
### Step 2: Substitute the simplified values into the expression
The expression now becomes:
[tex]\[ (2 \times 5i)(4 - 10i) \][/tex]
### Step 3: Perform the multiplication
Let's break it down into steps:
1. Compute [tex]\(2 \times 5i\)[/tex]:
[tex]\[ 2 \times 5i = 10i \][/tex]
2. Multiply [tex]\(10i\)[/tex] by [tex]\((4 - 10i)\)[/tex]:
- Use the distributive property to expand the multiplication.
[tex]\[ 10i \times (4 - 10i) = 10i \times 4 + 10i \times (-10i) \][/tex]
- Simplify each term:
- [tex]\(10i \times 4 = 40i\)[/tex]
- [tex]\(10i \times -10i = -100i^2\)[/tex]
Remember that [tex]\(i^2 = -1\)[/tex], so:
[tex]\( -100i^2 = -100 \times (-1) = 100 \)[/tex]
3. Combine the results:
[tex]\[ 40i + 100 \][/tex]
### Step 4: Write the final result in standard form
Express the final result as a complex number in the form [tex]\(a + bi\)[/tex]:
[tex]\[ 40i + 100 = 100 + 40i \][/tex]
Thus, the product of [tex]\((2 \cdot \sqrt{-25})(4 - \sqrt{-100})\)[/tex] is:
[tex]\[ 100 + 40i \][/tex]
