Math Problem Statement

Multiply (√(-16)) (√(-25)) and simplify your answer.

Solution

The expression you're being asked to simplify is:

(16)(25)\left( \sqrt{-16} \right) \left( \sqrt{-25} \right)

Here’s how to approach it:

  1. Simplify each square root:

    • 16=161=4i\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i (where ii is the imaginary unit, because 1=i\sqrt{-1} = i).
    • 25=251=5i\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i.
  2. Multiply the simplified results: (4i)(5i)=20i2(4i) \cdot (5i) = 20i^2

  3. Simplify further using the fact that i2=1i^2 = -1: 20i2=20(1)=2020i^2 = 20 \cdot (-1) = -20

So, the simplified answer is -20.

Would you like more details, or do you have any questions about this solution?

Here are 5 related questions to deepen understanding:

  1. What is the general rule for simplifying square roots of negative numbers?
  2. Why is i2i^2 defined as 1-1 in complex numbers?
  3. How would you multiply other complex numbers like (3+2i)(3 + 2i) and (4i)(4 - i)?
  4. What happens if you have square roots with mixed real and imaginary components, like 1625\sqrt{16} \cdot \sqrt{-25}?
  5. How do you graph complex numbers on the complex plane?

Tip: Always handle imaginary numbers carefully, making sure to simplify i2i^2 when multiplying them!

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Math Problem Analysis

Mathematical Concepts

Imaginary Numbers
Complex Numbers
Multiplication of Square Roots

Formulas

√(-a) = i√(a)
i² = -1

Theorems

Properties of Imaginary and Complex Numbers

Suitable Grade Level

Grades 9-12