Math Problem Statement
Solution
To fully simplify the expression , we need to handle the square roots of negative numbers, which involve imaginary numbers. Here’s a step-by-step simplification:
-
Rewrite the square roots of negative numbers: where is the imaginary unit, defined as .
-
Substitute and expand the expression:
-
Multiply the constants and imaginary terms: = -45 \cdot \sqrt{42 \cdot 6} \cdot (-1) \quad \text{(since i^2 = -1)}
-
Calculate the product under the square root:
-
Simplify :
-
Combine everything:
So, the fully simplified expression is:
Would you like further clarification on any step, or do you have additional questions?
Here are some related questions to explore further:
- What are imaginary numbers, and why do they involve ?
- How do you simplify expressions involving square roots of negative numbers?
- What is the process to simplify square roots like ?
- How would the expression change if it had additional imaginary terms?
- How can you apply similar steps to other expressions with complex numbers?
Tip: When dealing with square roots of negative numbers, always consider rewriting them in terms of (imaginary unit) to simplify further.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Complex Numbers
Imaginary Numbers
Simplification of Radical Expressions
Formulas
\( i = \sqrt{-1} \)
\( i^2 = -1 \)
Theorems
Properties of Imaginary and Complex Numbers
Suitable Grade Level
Grades 10-12
Related Recommendation
Simplify (4√2)(9√-7) Using Imaginary Numbers
Simplify Complex Expression 1 + √(-81) · √(-9) · √(-36)
Simplifying Expression Involving Square Roots of Negative Numbers and Imaginary Unit
Simplify √(-4) * √(-36) Using Complex Numbers
Multiplying Square Roots of Negative Numbers: Simplifying Complex Expressions