Math Problem Statement

Fully simplify \((\sqrt{-100})(\sqrt{5})\).

Solution

The given problem is to simplify (100)(5)(\sqrt{-100})(\sqrt{5}).

Step-by-Step Solution:

  1. Handle the negative square root: The square root of a negative number involves the imaginary unit ii, where i=1i = \sqrt{-1}. Therefore: 100=1001=10i\sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} = 10i

  2. Rewrite the expression: Substitute 100=10i\sqrt{-100} = 10i and 5\sqrt{5} remains as it is: (100)(5)=(10i)(5)(\sqrt{-100})(\sqrt{5}) = (10i)(\sqrt{5})

  3. Simplify the multiplication: Multiply the terms: (10i)(5)=105i(10i)(\sqrt{5}) = 10\sqrt{5}i

Final Answer:

105i10\sqrt{5}i

Would you like a detailed explanation of any specific step or have other questions?


Related Questions:

  1. What is the square root of a negative number and how does it involve the imaginary unit ii?
  2. How do you simplify expressions involving square roots and imaginary numbers?
  3. Can you explain why 1=i\sqrt{-1} = i is useful in complex numbers?
  4. What are some common rules for multiplying square roots in mathematics?
  5. How would you simplify (25)(12)(\sqrt{-25})(\sqrt{12})?

Tip:

When dealing with square roots of negative numbers, always factor out 1-1 and convert it to ii, then handle the remaining square root as usual!

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Math Problem Analysis

Mathematical Concepts

Imaginary Numbers
Square Roots
Simplification of Expressions

Formulas

Square root of a negative number: \( \sqrt{-x} = i\sqrt{x} \)
Product of square roots: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)

Theorems

Rules of Imaginary Numbers

Suitable Grade Level

Grades 9-12